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# COSTUME DESIGN THTR 25

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2539Coloquio39 2005516 page 1 a Surface subgroups and subgroup separability in 3manifold topology D D L ng UC Santa Barbara A W Reid UT Austin 6 9 25 coloqujo 2005516 page 2 469 Chapter 1 Introduction The picture that has emerged ofthe structure ofclosed 3manifolds over the last thirty years is that they are geometric which roughly speaking means that any 3manifold admits a canonical decomposi tions into pieces and each piece is in a certain sense which need not really concern us here modeled on one of eight geometriesi A rea sonable picture to bear in mind is the situation for closed surfaces in this case there are three models and any surface is either spherical at or hyperbolic depending on the sign of the Euler characteristici Of the eight geometries in dimension three all but the hyper bolic geometry are actually quite well understood but the structure of hyperbolic 3manifolds or indeed hyperbolic n manifolds remains rather mysteriousi This class has proved to be a magnet for research not only in topology but also in other elds including number the ory and geometric group theory This course will be concerned with aspects of the central question of whether hyperbolic 3manifolds al ways contain surface groups as well as the related but presumably deeper questions ofwhether such immersed surfaces can be promoted in a nite sheeted covering to an embedded surface Such questions can be approached in many ways here we shall introduce many of the classical facts related to such problems and in particular highlight the connection with group theory In this setting the surface group question is closely related to various separability properties of the fundamental groups of hyperbolic manifoldsi 3 i 2539coloquio39 2005516 page 3 4 CAP 1 INTRODUCTION These notes are organized as follows We begin with some in troductory material about the basics of hyperbolic manifolds The main consequence of the existence of a hyperbolic structure from this point of View is that one obtains a canonical representation of the fundamental group of the manifold into the group of isometries of the relevant hyperbolic space We then recall some classical facts about 3manifolds7 in particular we list four open conjectures which motivate all that follows In 37 we begin discussions of separability properties of groups and introduce the notions of residually nite and the much more powerful idea of subgroup separabilityi It turns out that these are intimately related to the questions about surface groups that we wish to pur sue7 and we show that these purely geometric notions have natural topological de nitions We go on to prove Scott7s theorem which was historically the rst real breakthrough in the area of subgroup sepa rabilityi We also describe certain surface groups which are known to be separable7 the socalled totally geodesic surfaces This leads naturally to the notion of an arithmetic hyperbolic 3 manifold and these are brie y explored one consequence of this tech nology is that one can show there are closed hyperbolic 3manifolds which contain no totally geodesic surface An important class of arithmetic manifolds is the socalled Bianchi groups and we sketch the recent proof that the Bianchi groups are subgroup separablei Finally we close with a discussion of new directions recently opened in the attacks on these problems in particular we discuss local re tractions over cyclic groups and show that Bianchi groups and most Coxeter groups admit such retractions 11 Preliminaries We begin with reviewing very brie y the notion of a hyperbolic n manifoldi We refer to 4 and 25 for standard facts about hyperbolic space7 many of which we will use without proof Since our focus here is largely algebraic7 many of the more basic geometric aspects can be easily ignored in this expositioni Let H denote the unique connected simply connected Rieman nian manifold all of sectional curvatures are 71 Suppose that M is 13 2539coloquio39 2005516 page 4 SEC 11 PRELIMINARIES 5 a closed topological manifold that is to say it is a compact Haus dorff and paracompactl topological space with the property that every point lies in an open neighbourhood homeomorphic to Eu clidean nspacel Then a hyperbolic structure on M is an atlas of charts 451 U A H with the property that on the overlaps we have VO E12 UUNV gt VU V CH is the restriction of a hyperbolic isometry from hyperbolic space to itself Such isometries are real analytic and a standard construction shows that a hyperbolic structure as de ned above produces a home omorphism DZMAH where M is the universal covering of Mr This identi cation in turn yields a way of transferring the canonical action of 7r1M on M to an action of 7r1M by isometries of H that is to say we obtain a faithful representation p 7r1M 7 lsomHn called the holzmomy representation As is discussed in the standard texts this representation has many properties the most important for us being that the action of p7r1 on H is properly discontinuous so the projection H A Hnrr1M is a covering map and we have a homemorphism M E H p7r1Ml The existence of a hyperbolic structure imposes many topologi cal constraints on the manifold For example the above discussion shows that M has contractible universal covering so that M is a K7r1 M 1 from which it follows for example that 7r1M is tor sion freer Perhaps the most important basic theorem about this sit uation is Mostow s rigidity theorem which says that for n 2 3 the representation p we have constructed above is the unique one up to conjugacy in IsamH with the properties of being discrete and faithfull This means that many geometrical properties of M for example its volume when equipped with a topological metric are actually topological invariants however we shall make scant use of these facts at least at the outset la 2539coloquio39 2005516 page 5 6 CAP 1 INTRODUCTION 111 Models of hyperbolic space and groups of isometries It will be convenient to recall two models of H that will be useful in what follows The upper half space model for H is de ned to be U zlzgulzn E R In gt 0 equipped with the metric de ned by 2 2 2 zlz2luzn I ds2 n This is particularly useful in low dimensions ilel n 23 since it affords a description of the groups of orientationpreserving isometries of these models as PSL2R and PSL2 C respectively The full groups of isometries of these models can be identi ed with PGL2 R and lt PSL2CT gt where 739 U A U is the isometry that is a re ection in the 11 zgplanel A description of the groups of isometries as linear groups is univer sally obtained by using the hyperboloid modell Let V Rn1 and equip V with the n1dimensional quadratic form lt 11 l l l 1 71 gt which we denote by fn throughout Consider the upper sheet of the hyperboloid fn 71 which we denote for now by Hnl Associated to the quadratic form fn is the bilinear form Bn V X V A R and this can be used to de ne a metric d H X H A R by decreeing that d is the function that assigns to each pair zy E H X H the unique number dz y such that coshdzy 7Bz H d de nes a metric space that is isometric to H with the metric described above and we henceforth we make no distinction The group of isometries of H can be identi ed as follows The orthogonal group of the form fn is On1R X E GLn1R X FnX Fn 2539coloquio39 2005516 page 6 a SEC 11 PRELIMINARIES 7 where Fn is the diagonal matrix associated to the quadratic form This is a lie group and has four connected components The subgroup of index 2 which preserves H is denoted 00n 1 R and is identi ed with lsomH Passing to the subgroup consisting of elements of determinant 1 denoted SOn1 R it follows that SOofn R may be identi ed with lsomJr 112 The nontriVial elements acting on H can divided into 3 classes parabolic elements elliptic elements and hyperbolic elements In terms of the action on H U SQ an element 7 is parabolic if it has precisely one xed point on Sgo l elliptic if it has no xed points on Sgo l but has a xed point in H and hyperbolic if it has two xed points on 551 In this last case if we denote the xed points by 1 and 1 there is a geodesic A7 called the axis of 7 in H with endpoints 1 and 1 and the element 7 acts by translating by some distance along A7 and possibly rotating through some angle In the case of n 23 one can make use of the description of elements as 2 X 2 matrices up to sign and get an algebraic char acterization In this setting a nontrivial element 7 is parabolic if tr2 7 4 elliptic if tr2 7 lt 4 and hyperbolic otherwise In the last case one can distinguish the class of purely hyperbolic elements ie those with tr2 7 gt 4 113 3Manifold topology and surface subgroups The case of most interest is when n 3 Closed hyperbolic 3 manifolds admit an amazing number of ways to be studied we shall take the point of View here which is not only of interest for itself but is connected to other mainstream problems in areas of mathematics other than topology A primary tool for understanding 3manifolds not at this stage necessarily hyperbolic emerged in the sixties in the seminal work 0 Waldhausen 32 To describe this work we need to introduce some de nitions for simplicity we shall restrict attention to the case of closed 3manifolds A closed 3manifold is said to be irreducible if every embedded 2sphere in M bounds a ball in M For example 1 2539coloquio39 2005516 page 7 8 CAP 1 INTRODUCTION the Schoenflies theorem shows that the 3sphere SS is irreducible and it is not hard to show that this implies that S39sF where F is some nite group acting freely is also irreducible In contrast if M is irreducible and has in nite fundamental group then one sees easily that its universal covering is contractible so that it is a K7r1M 1 for the purposes of these notes a good example to bear in mind is that of a closed hyperbolic 3manifold We say a closed 3manifold is su ciently large if there is an em bedding of a closed orientable 2manifold i F A M with the prop erty that either F is a 2sphere and the image of F does not bound a 3ball in M or F has genus at least one and the induced map ii 7r1 F A 7r1M is injective We often identify the surface with its image and refer to it as an incompressible surface in M Notice that this rules out the possibility of manifolds covered by 3sphere We also point out that with some fairly elementary 3manifold topol ogy if we assume that M is irreducible then it can be shown that an essential map M A 5391 can be homotoped so that a generic point preimage is incompressible and so in particular if H1 is in nite then it is sufficiently large Manifolds which are irreducible and sufficiently large are often referred to as Hakeri The results of 32 give a marvelous array of techniques for understanding Haken manifolds and a good deal more is known about this class than for the general irreducible 3 manifold even if this manifold has in nite fundamental group For example Waldhausen proves that such manifolds are determined up to homeomorphism by their fundamental groups and that homotopic homeomorphisms are in fact isotopic Historically speaking there was a feeling in the late 60s and early 707s that being Haken was rather common but the work of Thurston in the middle 707s suggested otherwise and nowadays it is felt that perhaps Haken 3manifolds are in some sense rather rare So one needs to develop techniques which suf ce even if the manifold is not Haken this being especially important for the case that the manifold is hyperbolic There has been an enormous amount of work in this direction in low dimensional topology we shall focus on just one aspect the notion of being virtually Haken and related ideas De nition 111 An irreducible 3mariifold M is de ned to be vir 25 39coloquio 392 2005515 page 8 SEC 11 PRELIMINARIES 9 tually Haken is there is a nite sheeted covering MF 8 M for which M F is Haken In contrast to the situation for Haken manifolds it s generally expected that every irreducible 3manifold with in nite fundamental group will in fact be virtually Haken and the key case in fact the only case left assuming the truth ofPerelman is that of a closed hyperbolic 3manifoldi In fact there is evidence that much stronger conjectures hold in particular all the following conjectures are generally expected to be true Surface Conjecture Every closed hyperbolic 3manifold contains the fundamental group of a closed orientable surface necessarily of genus gt 1 Conjecture 0 Every closed hyperbolic 3manifold has a nite sheeted covering which is Haken Conjecture 1 Every closed hyperbolic 3manifold has a nite sheeted covering which has H1M in nite Conjecture 00 Given a K gt 0 every closed hyperbolic 3manifold has a nite sheeted covering which has H1M in nite and of rank gt K These are of course ordered so that each conjecture implies all the conjectures above it The question of whether every closed hyperbolic 3manifold con tains a surface group has received a good deal of attention and while there are some results known 13 and 23 this is not the aspect of the problem that we shall concentrate on in these notes rather we shall ask the question suppose that one is given a surface subgroup how can it be used to address the other problems We note that a hyperbolic manifold is a KG 1 so standard obstruction theory guarantees that the inclusion map i 7r1 F A 7r1M is induced by a continuous map f F A M which we may suppose to be an immer sioni It is the desire to take this nonembedded 7r1 injective surface and the attempt to promote it to an embedding in a finite sheeted i 2539coloquio39 2005516 page 9 10 CAP 1 INTRODUCTION covering that guides us in what follows7 and relates to separability properties of groups 114 An example A good example of a Haken hyperbolic 3manifold to keep in mind is that of a surface bundle over the circle These are constructed as follows Let 2g be a closed orientable surface of genus g 2 2 and 45 g A By a homeomorphismi The Mapping Torus of 45 is the closed 3manifold7 denoted M that is formed by taking 39 X 071lI70E I71A From this description7 M is seen to be a ber bundle over 5391 and the bers are embedded incompressible surfaces homeomorphic to Egi In particular these manifolds are Haken indeed the rst Betti number of M is positive Part of proof Thurston7s hyperbolization theorem shows that M is hyperbolic if and only if 45 is a pseudoAnosov map This construc tion ts with the conjectures discussed above A strengthening of Conjecture 1 above asks Conjecture 1 Let M be a closed hyperbolic 3manif0ld Then M has a nite sheeted cover which is a surface bundle over the circle 2539coloquio39 2005516 page 10 Chapter 2 Separability properties of groups Motivated by the discussion in 32 our point of View is that one wishes to try and understand a complicated in nite group7 namely 7r1M7 and one way of doing this is to attempt to understand the nite quotients of this group equivalently7 the standard theory of covering spaces shows that one might wish to gain insights into M by thinking about its nite covering spaces This raises an immediate problem On the face of it7 there is no a priori reason to think that the group 7r1M has any subgroups of nite index at all This is the rst issue that we shall address and is the motivation for the next section 21 Residual niteness De nition 211 A group G is said to be residually nite given any nonidentity element g E G there is a subgroup of nite index H in G with g H By using the action of G on the left cosets of H by left translation7 we obtain a permutational representation p G SGH from which it follows that H contains the normal subgroup of nite index kerp 11 i 2539coloquio39 2005516 page 11 12 CAP 2 SEPARABILITY PROPERTIES OF GROUPS and there is therefore a homomorphism 45 G a A Gkerp where it is visible that lAl lt 00 and f 1 Conversely if such a homomorphism exists then g kerq which has nite index so that these conditions are equivalent We shall use them both We also note that the restriction to one element is not necessary one sees easily that it is equivalent replace the element g by any finite set of elements in the above definition Residual finiteness guarantees a large supply of subgroups of finite index in G indeed it shows that the intersection of all the subgroups of finite index in G yields only the identity element 1 For future reference we notice that the algebraic condition above is equivalent to the following geometric condition which we do not attempt to state in the most general form possible Lemma 212 Suppose that M is a closed manifold and G 7r1M Then G is residually nite if and only if the following condition holds For every compact subset C ofM there is a nite sheeted covering MF ofM so that the natural map M A MF is an embedding when restricted to C Proof We recall that the natural action of 7r1M on its universal covering is properly discontinuous that is to say given any compact set C the number of group elements for which gC O O Z is finite If we assume the group G is residually finite we can find a sub group of finite index H in G which excludes this finite number of group elements and elementary covering space theory now shows that the set C embeds in the finite covering MF corresponding to H Conversely suppose the geometric condition holds and we are given a nontrivial element of the fundamental group g We may represent g by a based map 01 A M and nontriviality is equivalent to the preimage of this map being an arc in the universal covering which is to say the endpoints are distinct Taking these two endpoints as the compact set C we see that there is a finite sheeted covering of M in which this arc fails to close up that is to say a finite sheeted covering MF to which the loop g does not lift as a loop Covering space theory shows that the element g does not lie in the subgroup 2539coloquio39 2005516 page 12 SEC 21 RESIDUAL FINITENESS 13 corresponding to the covering Mp It will also be useful to note the following simple group theoretic facts Lemma 213 Let G be a group and H a subgroup ofG If G is residually nite then so is H If H is residually nite and G H lt 00 then G is residually nite Proof Take any nonidentity element h of H G is residually nite so there is a homomorphism to a nite group 45 G A A which does not kill g restrict this homomorphism to H ii Given a g E G either g H in which case we are done7 or we can nd a subgroup of nite index in H which excludes g this subgroup also has nite index in G While it is by no means true that all groups are residually nite see the example given at the end of this section7 and indeed there are in nite groups with no subgroups of nite index at all7 see for example 287 many of the groups which arise in nature are7 and7 in fact7 this is a fairly soft property in the sense that there are quite general results which guarantee that a group is residually nite The most famous of these7 and the most useful for us7 is Mal cev7s theorem Theorem 214 Let R be a nitely generated integral domain Then for any nonidentity element g E GLn7 R there is a nite eld K and a homomorphism qt GLnR A GLn7 K so that f 1 Proof The key ingredient is the following purely algebraic result Lemma 215 Let R be a nitely generated integral domain Then The intersection of all the maximal ideals ofR is the zero ideal IfVl is any maximal ideal of R then RM is a nite eld If we assume this result7 we may prove Mal cev7s theorem The given element g is not the identity element7 so that at least one of the elements of the difference g 7 I is nonzero x such an element la 2539coloquio39 2005516 page 13 14 CAP 2 SEPARABILITY PROPERTIES OF GROUPS an denote it by r By of the Lemma7 there is a maximal ideal M which does not contain r and by ii the quotient RM is a nite eld The map induced from projection 45 GLn7 R GLnRM has the required properties7 since by choice of r7 the element is not the same in the quotient as the element 45 I This result is used in the following fashion As discussed in 3l7 the groups lsomH are all subgroups of GLn 17R If M is a hyperbolic nmanifold7 then although the canonical representation p 7r1M A lsomH lt GLnl7 R apparently takes its values in the real field7 we can form a nitely generated integral domain by taking a generating set for 7r1 and looking at the subring ofR generated by the entries of the p images of these generators together with 1 say Denoting this ring by R we see that p 7r1M A GLn 17R Since p is injective we have proved Theorem 216 The fundamental group of a closed hyperbolic n manifold is residually nite lndeed7 MallceV7s theorem is a rich source of residually nite groups7 since many commonly occurring groups have faithful linear representations7 free groups and surface groups being the most obVi ous examples Assuming Perelman s solution to geometrization in dimension 3 it follows that all compact 3manifolds have residually nite fundamen tal groups lndeed7 for Haken manifolds this can be established with out geometrization see 157 and the proof assuming geometrization builds on 15 However7 the following is still an interesting open problem Question 217 Let M be a Haken 3manifold Does 7r1M admits a faithful representation into GLn7 C for some n Example of a nonresidually nite group The most famous class of examples of nitely generated nonresidually l3 2539coloquio39 2005516 page 14 SEC 22 SUBGROUP SEPARABILITY 15 nite groups are the BaumslagSolitar groups de ned as follows Let p and g be natural numbers and de ne the group BSpq lt a7bl bapb 1 a 1 gt If neither p or q is 17 it was shown in 3 that the group BS p7 q is not residually nite Note that when p 1 the group BS1q is linear and hence residually nite The linear representation is described as follows take a 1 gt and a 0 0 1 0 1Lj 22 Subgroup separability We now introduce a property7 that while being related to residual niteness is a good deal stronger De nition 221 Let G be a group and H a nitely generated sub group G is called Hsubgroup separable given any g E GH there exists a subgroup K lt C of nite index with H lt K and g K G is called subgroup separable or LERF if G is Hsubgroup separable for all nitely generated H lt C In contrast to the situation for residually nite groups7 one cannot reduce to the case that K is normal in G since conjugates of K will not in general contain H It is left to the reader to verify that the equivalent condition which refers to homomorphisms in this case is The subgroup H separable in the group G if and only if for element g H7 there is a homomorphism qt G A A where A lt 00 and It is also left to the reader to verify that the following group theoretic properties continue to ho Lemma 222 Let G be a group and H a subgroup ofG If G is subgroup separable then so is H IfH is subgroup separable and G H lt 00 then G is subgroup separable 2539coloquio39 2005516 page 15 a 16 CAP 2 SEPARABILITY PROPERTIES OF GROUPS Subgroup separability is an extremely powerful property and it is much stronger than residual nitenessi In particular there is no theorem analogous to Mallcev7s more or less every example must be treated on its individual meritsi The class of groups for which subgroup separability is known for all nitely generated subgroups is extremely small abelian groups free groups surface groups and carefully controlled amalgamations of these Exam les i Hall 14 Free groups are subgroup separablei ii Scott 27 Surface groups are separablei iii Folklore If A and B are subgroup separable then so is A 96 Bi iv If A B and C are nite then A 0 B is subgroup separablei It contains a free subgroup of nite index 26 The powerful nature of this property is underlined by listing some apparently wellbehaved groups which fail to be subgroup separablei We note that SL2 Z contains a free subgroup of nite index so that it follows from Hall s theorem together with iv above that it is sub group separablei However as we discuss in 5 this fails for SLn Z when n 2 3 Another interesting example is the following Let Fn denote the free group of rank n Then Fn gtlt Fn is not subgroup sepa rable although it is the direct product of subgroup separable groups This failure can be attributed to the lack of a solution to the general ized word problem for these groups see 21 for instance It is known that LERF like residual niteness for the word problem implies a solution to the generalized word problemi The groups BSlq are residually nite but not LERFi This can be seen by checking that the the cyclic subgroup lt a gt is not separablei This is a special case of a more general result of Blass and Neumann 5 that shows that if G is a group and H lt C with the property that H is conjugate into a proper subgroup of itself then G is not Hseparablei From the point of view of 3dimensional topology it is known that there compact 3manifolds whose fundamental groups are not LERFi These examples are all closely related to an example in 11 of a lpunctured torus bundle over the circle whose fundamental group is not LERFi 25 39coloquio 392 2005515 page 16 SEC 22 SUBGROUP SEPARABILITY 17 There is also an analogue of a geometric equivalence Lemma 223 Suppose that M is a closed manifold and G 7r1 Then G is subgroup separable and only the following condition holds For every nitely generated subgroup H lt 7r1 and every com pact subset C of MH there is a nite sheeted covering MF MK A M subordinate to MH ie H S K so that the nat ural map A is an embedding when restricted to C It is this geometric equivalence Which fuels much of the interest in this property in low dimensional topology and is discussed in detail in the next section 2539coloquio39 2005516 page 17 Chapter 3 Subgroup separability and Scott s theorem As described in 213 one of the central problems we are interested in solving is that if we are given an immersed 7r1injective surface F can we promote it to an embedding in a nite sheeted covering of the ambient 3manifold Mi Restricting attention to the hyperbolic case it is known that the manifold H37r1 F is a topological product F X R and therefore contains an embedding of closed orientable surface homeomorphic to Fr Taking this surface to be the compact set C in the geometric version of the subgroup separability property described by Lemma 2213 we see that there is a subgroup of nite index K gt 7r1F in 7r1 M so that the surface F embeds in HSKi There is an important reduction which we introduce at this point We rst need a de nition We refer to 25 for some of the details that we omit Suppose that H is a subgroup of a discrete group of hyperbolic isometries Ci For technical reasons we need to exclude subgroups H which are very small in the sense that it contains a soluble subgroup of nite index Then associated to H is a canonical set its limit set which we de ne as the closure in the sphere at in nity of hyperbolic space of the union of all the xed points of hyperbolic elements of Hi We denote the limit set by It is clear from this 18 i 2539coloquio39 2005516 page 18 19 de nition that the limit set is H invariant and provided we exclude small subgroups it contains in nitely many points We can use this set to construct an H invariant subset lying inside H To this end we de ne a totally geodesic hyperplane to be any codimension one totally geodesic submanifold of H The closed set lying to one side of a totally geodesic hyperplane is a closed half space We de ne the convex hull of the limit set CAH to be the intersection of all those half spaces which contain the limit set One can see easily from the de nition that the convex hull is the smallest convex closed set which contains all the geodesic axes of elements of H The set CAH is visibly H invariant and we may form the quotient CA One de nes H to be geometrically nite if this set is compact Or nite volume in the case that H is a subgroup of a nite volume hyperbolic group If H is not geometrically nite it is called geometrically in nite It turns out that in the case of subgroups of lsomH2 that geo metrically nite is equivalent to nitely generated but in general the situation is more complicated It is not entirely elementary but not hard that if H is a normal subgroup of G then AH AG In particular in the case of a hyperbolic surface bundle over the circle as described in 24 the limit set of the bre surface is the same as the limit set of the whole 3manifold group ie the whole 2sphere at in nity Somewhat amazingly the separability situation in this apparently more complicated context can actually be resolved The recent so lution of the Tameness conjecture by Agol 1 and independently by Calegari and Gabai 12 shows that any nitely generated geometri cally in nite subgroup A of the fundamental group of a nite volume hyperbolic 3manifold M is a virtual ber that is to say M has a nite sheeted cover that is a hyperbolic surface bundle over the circle and the ber group is A Combining this with prior work of Thurston and Bonahon 29 and 6 we summarize what is important for us in the following theorem Theorem 304 Suppose that M is a nite volume hyperbolic 3 manifold Then the nitely generated geometrically in nite subgroups of 7r1M are separable in 7r1M It follows from this result that we may restrict attention to the 3 2539coloquio39 2005516 page 19 20 CAP 3 SUBGROUP SEPARABILITY AND SCOTT39S THEOREM geometrically nite surface groups and indeed guided by this and other considerations the general direction of the theory has been trying to separate geometrically nite subgroups or in the case of negatively curved groups in the sense of Gromov the quasiconvex subgroups The results of Hall and Scott t into this pattern since coincidentally in those settings nitely generated and geometrically nite turn out to be equivalent 31 Scott s theorem The proof that free groups are subgroup separable given by Hall was essentially algebraic but can be made geometric fairly easily How ever little progress was made for many years until a new technique was introduced by Scott 27 We shall now give an exposition of this technique recast in modern language and somewhat re ned as described in Suppose that P is a nite volume polyhedron in H all of whose dihedral angles are 7r2 Henceforth we call this an all right polyhe dron Then the Poincare polyhedron theorem implies that the group generated by re ections in the codimension one faces of P is discrete and a fundamental domain for its action is the polyhedron P that is to say we obtain a tiling of hyperbolic nspace by tiles all isometric to P Let the group so generated be denoted by CP Theorem 311 The group CP is H subgroup separable for every nitely generated geometrically nite subgroup H lt CP Proof As in Lemma 223 the separability of H is equivalent to the following Suppose that we are given a compact subset X C Then there is a nite index subgroup K lt CP with H lt K and with the projection map 4 HnH HnK being an embedding on X e sum up the strategy which achieves this geometric condition The group H is geometrically nite and one can enlarge its convex hull in HnH so as to include the compact set X in a convex set contained in HnH this convex set lifts to an Hinvariant convex set inside H One then de nes a coarser convex hull using only 2539coloquio39 2005516 page 20 SEC 3 1 SCOTT39S THEOREM 21 the hyperbolic halfspaces bounded by totally geodesic planes which come from the P tiling of H this hull is denoted by HpCi This hull is Hinvariant and the key point is to show that Hp CH only involves a nite number of tiles The remainder of the proof follows 27 and is an elementary argument using the Poincare polyhedron theorem and some covering space theory We now give the details Let C be very small neighbourhood of the convex hull of H7 re garded as a subset of Hni In our setting7 the group CP contains no parabolic elements so that the hypothesis implies that CH is compact The given set X is compact so that there is a t with the property that every point of X lies within a distance t of CHi Let OJr be the lOt neighbourhood of C in Hni This is still a convex Hinvariant set and CurH is a compact convex set containing Xi As discussed above7 take the convex hull Hp 6 of OJr in H using the half spaces coming from the Ptiling of Hni By construction Hp CJV is a union of P tiles7 is convex and Hinvarianti The crucial claim is Claim Hp CH involves only a nite number of such tilesi To see this we argue as follows Fix once and for all a point in the interior of a top dimensional face of the tile and call this its barycentre The tiles we use actually often have a geometric barycentre ie a point which is equidistant from all of the faces but such special geometric properties are not used it is just a convenient reference point Our initial claim is that if the barycentre of a tile is too far away from 07 then it cannot lie in Hp Ci The reason for this is the convexity of C3 If a is a point in H not lying in CJr then there is a unique point on CJr which is closest to a Moreover7 if this distance is R then the set of points distance precisely R from a is a sphere touching OJr at a single point p on the frontier of 04F and the geodesic hyperplane tangent to the sphere at this point is the generically unique supporting hyperplane separating OJr from a Suppose then that P is a tile whose barycentre is very distant from Cf Let a be the point of P which is closest to 04F and let 2539coloquio39 2005516 page 21 22 CAP 3 SUBGROUP SEPARABILITY AND SCOTT39S THEOREM p be a point on the frontier of CJr which is closest to P As noted above there is a geodesic supporting hyperplane Hp through p which is generically tangent to 04F and separates OJr from a Let the geodesic joining a and p be denoted by 7 Note that since p is the point of OJr closest to a 7 is orthogonal to Hpi If a happens to be in the interior of a tile face of P then this tile face must be at right angles to 7 since a was closesti Let Hat be the tiling plane de ned by this tile facei Since in this case 7 is at right angles to both HW and Hp these planes are disjoint and so the tiling plane separates P from OJr as required If a is in the interior of some smaller dimensional face a then the codimension one faces of P which are incident at 0 cannot all make small angles with 7 since they make right angles with each other The hyperplane H which makes an angle close to 7r2 plays the role of Ha in the previous paragraph The reason is that since a and p are very distant and the planes Hp and H both make angles with 7 which are close to 7r2 the planes are disjoint and we see as above that P cannot lie in the tiling hull in this case either The proof of the claim now follows as there can be only nitely many barycentres near to any compact subset of H Hi The proof of subgroup separability now nishes off as in 27 Let K1 be the subgroup of CP generated by re ections in the sides of HPCi The Poincare polyhedron theorem implies that HPC is a noncompact fundamental domain for the action of the subgroup KL Set K to be the subgroup of CP generated by K1 and H then HnK Hp CH so that K has nite index in CP Moreover the set X embeds as required 32 Totally geodesic surfaces As described above one of the interests for lowdimensional topol ogy in the subgroup separability condition is to pass from a surface subgroup to an embedded nonseparating surface in a nite sheeted coveringi However this makes clear that we have no real need to sep arate all nitely generated subgroups certain special classes suf ce 2539coloquio39 2005516 page 22 SEC 3 2 TOTALLY GEODESIC SURFACES 23 In particular7 there is a very restricted class of subgroup which can always be separate De nition 321 A closed surface group 7r1F lt 7r1M in a closed hyperbolic manifold is said to be totally geodesic if the discrete faithful representation of 7r1M can be conjugated so that the image of 7r1F lies inside PSL2R We have stated this condition algebraically since it is in this form that we shall use it7 but it has a natural interpretation in the context of differential geometry As usual we can construct an immersion i F lt gt M realising the surface group and the totally geodesic condition means that the metric induced on the surface F from the ambient 3manifold M can be arranged to be a constant curvature hyperbolic metric In this sense they are at the opposite end of the spectrum from geometrically in nite surfaces In the language introduced above7 the universal covering of F is a totally geodesic hyperplane in H3 The importance of totally geodesic surfaces for us is the following theorem Theorem 322 Let M be a closed hyperbolic 3manifold containing a closed totally geodesic surface F Then there is a nite sheeted covering of M which contains an embedded closed orientable totally geodesic surface To prove this theorem7 we rst establish Lemma 323 Let C be a circle or straight line in C U 00 and M HSlquot a closed hyperbolic 3manifold Let stabClquot 7 E F 7C C Then stabC7 F is separable in F Proof Let H denote stabClquot We may assume without loss of generality that H is nontrivial since F is residually nite Note that H is either a Fuchsian group or a Zgextension of a Fuchsian group To prove the Lemma we need to show that given g H then there is a nite index subgroup of F containing H but not g By conju gating7 if necessary7 we can assume that H stabilizes the real line Denoting complex conjugation by 7397 then 739 extends to SL27 C and 2539coloquio39 2005516 page 23 24 CAP 3 SUBGROUP SEPARABILITY AND SCOTT39S THEOREM is wellde ned on PSL2 C The stabilizer of R in PSL2 C is then characterized as those elements 7 such that 77 7 The rest of the argument is similar in spirit to the proof of Mal cevls theoremi Let F be generated by matrices 9192 i i i 9 Let R be the subring of C generated by all the entries of the matrices 9139 their complex con jugates and 1 Then R is a nitely generated integral domain with 1 so that for any nonzero element there is a maximal ideal which does not contain that element Note that both T and 7F embed in PSL2 R If 7 E F H then 7 Pg where at least one element from each set 7gij igij739gijgij ij 12 is nonzero Call these zy and choose a maximal ideal M such that my Mi Let p PSL2R A PSL2RM gtlt PSL2RM be the homomorphism de ned by My 7r77r7397 where 7r is induced by the natural projection R A RMi e image group is nite since RM is a nite eld By construction the image of 7 is a pair of distinct elements in PSL2RM while the image of H lies in the diagonal This proves the lemma Proof of Theorem 322 Let 239 F A M be a totally geodesic immersion of a closed surface Let F be the covering group of M in PSL2 C and let H i7r1Fi Then H is Fuchsian and preserves some circle or straight line C in CUooi Notice that stabC F contains H and is a discrete group acting on the hyperbolic plane spanned by C therefore we have a nite sheeted covering HQH A HQstabC T which implies that stabClquot is the fundamental group of a closed orientable surfacei Now by Lemma 323 the group stabClquot is separable in F and there is an embedded closed surface in the covering H2stabClquoti As discussed at the start of 4 separability now implies that there is a nite sheeted covering MK in which this surface embedsi By un twisting this surface if it happens to be nonorientable it follows that this manifold or its double covering contains the required embedded orientable totally geodesic surface In fact one can go further and produce in nite virtual Betti number in this case la 25 39coloquio 392 2005515 page 24 SEC 33 25 Theorem 324 If a closed hyperbolic 3manif0ld M contains a to tally geodesic closed surface group then it has in nite virtual Betti number Proof As we showed above7 the manifold virtually contains an em bedded totally geodesic surface7 F Suppose that this surface is sep arating7 then it expresses the fundamental group of the manifold as a free product with amalgamation 7r1 E 7r1L amp 7r1R where L and R are the two sides By untwisting if necessary7 we may sup pose that the indices n1L 7r1F and n1R 7r1F are both in nite Now separability guarantees that we can nd a homomor phism of 45 7r1M A A onto a nite group A so that the indices 457r1L 457r1F and W1R 457r1F are both strictly larger than two By restricting this homomorphism to both sides7 we assemble a new map W1M 4157r1 D 457r1F 4157r1 R onto a free product with amalgamation of nite groups Now as in 26 the target group here is virtually a free group of rank two or greater so that 7r1M has in nite virtual Betti number as required Remark Theorems 322 and 324 and Lemma 323 were stated only for closed hyperbolic 3manifolds However7 these results also hold7 and are proved the same way for noncompact nite volume hyperbolic 3manifolds 33 In light of Theorem 324 and Conjectures 07 1 and 00 from 27 a nat ural question is whether every nite volume hyperbolic 3manifold contains an immersion of a closed totally geodesic surface The an swer to this is no7 but to discuss this in more detail we will require some arithmetic properties of hyperbolic 3manifold groups7 and we shall develop some of this below see 24 for more on this topic Let F be a subgroup of PSLQ7 C that does not contain a soluble subgroup of nite index The trace eld of lquot7 denoted QtrF is the eld Qtr7 7 E lquot i 2539coloquio39 2005516 page 25 26 CAP 3 SUBGROUP SEPARABILITY AND SCOTT39S THEOREM Note that7 for any 7 E PSLQ7 C7 the traces of any lifts to SL27 C will only differ by i and so the trace eld is wellde ned The trace eld is a conjugacy invariant of lquot The connection to nite volume 3 manifolds is made interesting by the following result that is basically a consequence of Mostow Rigidity in dimension 3 Theorem 331 Let F lt PSL2C be a discrete group of nite covolume then QtrF is a nite extension of Q It follows that if M HSlquot is a hyperbolic 3manifold of nite volume7 then QtrF is a topological invariant of M ne de ciency of the trace eld is that it is not a commensura bility invariant7 however7 this is easily remedied De nition 332 Let F be a nitely generated group De ne Fa lt 72 l 7 E F gt Then it is easy to check that Ta is a nite index normal subgroup of F whose quotient is an elementary abelian 2group Theorem 333 Let F be a nitely generated nonelementary sub group of SL2C The eld Qtrlquot2 is an invariant of the com mensurability class of F Notation The eld Qtrlquot2 is denoted klquot and is called the invari ant trace eld of 1quot Another algebraic object that plays a role in the theory of hyperbolic 3manifolds is a quaternion algebra over the invariant trace eld Suppose F is a subgroup of PSLQ7 C that does not contain a soluble subgroup of nite index in fact we should work in SL27 Here we associate to F a quaternion algebra over Qtrlquot Let AoF 3am l az E QtrF739 E Flv where only nitely many of the ai are nonzero Theorem 334 A01quot is a quaternion algebra over QtrF In the case of Agra we denote this by AT and call this the in variant quaternion algebra of lquot 2539coloquio39 2005516 page 26 a SEC 34 27 Example Suppose that HSlquot is noncompact but nite volume eg the gureeight knot complement Then AF M27 klquot The only thing to note is that the manifold being nite volume and non compact implies that F contains parabolic elements These give rise to zero divisors in AF and hence the invariant quaternion algebra is not a division algebra The isomorphism follows Notation A quaternion algebra B over a eld h of characteristic 2 can be described as follows Let a7 b E 16 and let B be the 4 dimensional vector space over h with basis 17i7j7k Multiplication is de ned on B by requiring that l is a multiplicative identity element7 that 12 ai j2b1 ij 7jik 31 and extending the multiplication linearly so that B is an associative algebra over h This algebra is denoted by the Hilbert symbol 3 34 We now produce manifolds without totally geodesics surfaces This will follow from our next theorem which requires one more de nition Let B be a quaternion algebra over a number e d h We say that B is rami ed at an embedding a h A C if 006 C R and the quaternion algebra B0 006 R g D where D is the Hamiltonian quaternions7 and B is the quaternion algebra over 0h obtained by applying a is a prime ideal of Rk the ring of integers of h we say that a quaternion algebra Bh is rami ed at 73 if B k hp is a division algebra over the local eld hp ie the completion of h at 73 Theorem 341 Let F be a Kleinian group of nite covolume which satis es the following conditions 0 klquot contains no proper sub eld other than Q 0 AF is rami ed at at least one embedding of klquot Then T contains no purely hyperbolic elements 2539coloquio39 2005516 page 27 28 CAP 3 SUBGROUP SEPARABILITY AND SCOTT39S THEOREM Before discussing the proof we recall that a purely hyperbolic el ement is a hyperbolic element 7 with tr27 gt 4 Since a Fuchsian group is conjugate to a subgroup of PSL2 R the generic element is purely hyperbolici More precisely if F is a cocompact Fuchsian group then all elements of in nite order are purely hyperbolici Thus Theorem 341 provides a method of checking the lack of existence in a very strong way of totally geodesic surface subgroupsi Proof Note that 1quot contains a hyperbolic element if and only if lquot2 contains a hyperbolic elementi Let us suppose that 7 E lquot2 is hy perbolic and let t tr7i By assumption t E klquot N R Q and M gt 2 Now Alquot is rami ed at an embedding a of klquot which is necessarily reali Let a klquot A R be the Galois embedding of klquot and 1 A1quot A D extending 0 Thus 1mm c AF1C Dli But then since t E Q t W ww a 1W W mm Since tr D1 C 722 we obtain a contradiction D Now although it may appear the conditions of Theorem 341 are hard to check many examples of manifolds are known to satisfy these conditions This is the class of arithmetic hyperbolic 3manif0lds see 24 for details These are de ned as follows Let k be a number eld with exactly one pair of complex conjugate embeddings let Rk denote the ring of integers of k and let B be a quaternion algebra over k which is rami ed at all the real embeddings Let p be a kembedding of B into M2 C and let 0 be an Rkorder of Bi Then a subgroup lquot of PSL2 C is an arithmetic Kleinizm group if it is commensurable with some such PMOI Examples of arithmetic Kleinian groups that satisfy the hypoth esis of Theorem 341 are then easily constructedi One just takes for example a cubic number eld and B a quaternion algebra with the properties given in the de nition A particular example of such 2539coloquio39 2005516 page 28 SEC 35 29 a group manifold is the Weeks manifold7 the hyperbolic 3manifold with the smallest known volume at this point The e Qlt6gt where 9 is a complex root of 13 7 z 1 07 and B is a quaternion algebra over h rami ed at the real embedding of h and at a prime ideal of norm 5 in h 35 An important subclass of arithmetic Kleinian groups are the Biahchi groups These are the generalization to dimension 3 of the modular group and are de ned as PSLQ7 00 where 00 is the ring of integers in These groups all contain a copy of PSLQ7 Z and so all contain a nonelementary Fuchsian group However7 they also contain lots of cocompact Fuchsian subgroups These are constructed as follows Lemma 351 Let F be a Fuchsizm subgroup of the Biahchi group PSL20d which contains two honcommuting hyperbolic elements Then F preserves a circle or straightline in C U 00 alzl2BzEc07 where a706 Z 1th 6 00 Proof Since F is a Fuchsian subgroup7 it does preserve a circle or straightline C in C U 00 Assume this has equation ale B2 70 0 with a and 5 real numbers and B complex By conjugating in PSLQ7 00 we may assume that a f 0 Hence on further dividing7 we can assume that a 1 F contains a pair of noncommuting hyperbolic elements7 and these have distinct xed points which lie on C recall 22 If one a 5 6 such element g is represented by then its xed points are D Yi where A2 a 52 7 4 gt 0 An easy calculation shows that the perpendicular bisector of the line in C joining these xed points has the equation 72WWW7 2539coloquio39 2005516 page 29 30 CAP 3 SUBGROUP SEPARABILITY AND SCOTT39S THEOREM where M 27 Since the centre of C is the intersection of two such lines we deduce since all these coef cients are in Qx7d that B E Q 7d Rewriting the equation of the circle C as l2 l2 lBl2 67 we nd that since the xed points of the hyperbolic element 9 lie on C we can solve for c E Q Clearing denominators completes the proof of the lemma D Using arithmetic methods it can be shown that if C is a circle or straight line as above and Hg the hyperbolic plane in H3 erected on C then stabC PSL2 001 if a Fuchsian group acting with nite coarea and indeed one can arrange that the group acts cocompactlyi Example In the case ofd l the Bianchi group is the Picard group PSL2 If we let p E Z be a prime congruent to 3 mod 4 then the circle Cp2 Czlzl2p gives rise to a subgroup stabCpPSL2 that is cocompacti 36 The discussion in 4i374i5 exploits the description of lsomH3 as PSL2 C In higher dimensions since lsomH 00n 1 arith metic methods of constructing lattices exploit the theory of quadratic forms We now discuss this construction This goes back to Borel and Harish Chandra 8 and and will require some standard facts about quadratic forms and orthogonal groups of such forms 17 is a standard reference 37 If f is a quadratic form in n 1 variables with coef cients in k and associated symmetric matrix F let 0fXEGLn1C XtFXF 2539coloquio39 2005516 page 30 SEC 37 31 be the Orthogonal group of f7 and SOU 00 7 SM 17C7 the Special Orthogonal group of These are algebraic groups de ned over kl De nition 371 Two ndimensional quadratic forms f and L de ned over a eld k with associated symmetric matrices F and are equivalent over k there exists P E GLn7 K with P FP If k C R is a number field7 and Rk its ring of integers7 then SOfRk is an arithmetic subgroup of SOfR7 8 or The following is wellknown and proved in 2 for example Lemma 372 Let k C R be a number eld and Rk its ring of inte gers Let f and L be ndimensional quadratic forms with coe cients in RK which are equivalent over 16 o SOf R is conjugate to SOq R and S00 h is conjugate to 304 k 0 SOf Rk is conjugate to a subgroup of SO q k commensurable with SOqRk D There is a converse to the second part of Lemma 372 Which we record here see 31 for example Note that if f Af for A E k nonzero7 then SOON h SOf k With notation as above7 Lemma 373 Suppose SOfRk and SOqRk are commensu rable Then f is equivalent to Aq for some nonzero A E K Assume that k C R is totally real7 and let f be a form in n lvariables With coef cients in h an be equivalent over R to the form Furthermore7 if a z k A R is a eld embedding7 then the form f obtained by applying a to f is de ned over the real number eld 03906 We insist that for embeddings a id7 f is equivalent over R to the form in n ldimensions7 of signature n 10 Since f is equivalent over R to fn7 it from follows Lemma 372 that Of R is conjugate7 by a matrix P say in GLn 17 R to 0fn R From 8 or PSOO 133 1 de nes an arithmetic subgroup in i 2539coloquio39 2005516 page 31 32 CAP 3 SUBGROUP SEPARABILITY AND SCOTT39S THEOREM lsomH and so necessarily of nite covolume In what follows we will abuse notation and suppress the conjugating matrix and simply identify S000 Rk as an arithmetic subgroup of lsomJr The group S000 Rk is cocompact if and only if the form f does not represent 0 nontrivially with values in h see Whenever n 2 4 the arithmetic groups constructed above are noncocompact if and only if the form has rational coef cients since it is well known every inde nite quadratic form over Q in at least 5 variables represents 0 nontrivially see 17 he following theorem summarizes a case that what we shall make use of see 31 Chapter 6 Theorem 374 If F is a noncocompact arithmetic subgroup of SOofn R then T is commensurable up to conjugacy with a group S000 Z where f is a diagonal quadratic form with rational coe cients and signature n1 D 38 We will now discuss the proof of the following theorem This was proved in 2 for geometrically nite subgroups Since appearing the solution to the Tameness conjecture and 12 has allowed for the extension to LERF Theorem 381 The groups PSL2 0d are LERF Proof We rst remark that if H is a nitely generated subgroup of PSL2 00 it contains a torsionfree subgroup of nite index and so if F is geometrically in nite then it will be separable by Theorem 304 and the fact that a nite index supergroup is separable Thus it remains to show that PSL 2 00 is Hseparable for H a geometrically nite subgroup he key geometric idea is contained in the following lemma Lemma 382 For every d there is a nite index subgroup Ad of PSL2 00 such that A is contained in the group generated by reflec tions in a nite volume all right polyhedron P in H5 Given this Theorem 381 follows immediately from Theorem 311 13 2539coloquio39 2005516 page 32 SEC 38 33 Sketch of the Proof of Lemma 382 This makes use of the description of arithmetic groups arising from quadratic forms Theorem 374 affords a description of the non cocompact lattices and it can be shown that if pd is the quadratic form lt dll7l gt then pd is isotropic and so SOopdZ is a noncocompact arithmetic group Indeed it represents the commen surability class as discussed above of the image of PSL2 00 in SOO31 We now discuss the construction of the all right polyhedron a related arithmetic group and the group Ad An all right ideal polyhedron in hyperbolic 6space In H6 there is a simplex E with one ideal vertex given by the following Coxeter diagram see 25 p 301 4 Figure 1 Notice that deleting the right most vertex of this Coxeter symbol gives an irreducible diagram for a nite Coxeter group namely E5 This group has order 27345 The connection to arithmetic groups is given in the following lemma Lemma 383 OWE SOOUB Z There is an all right polyhedron Q built from 27345 copies of E In particular the re ection group is commensurable with S00 f5 Z Proof The rst part is due to Vinberg 30 and also discussed in 25 p 301 For the second part as noted above if one deletes the face F of the hyperbolic simplex corresponding to the right hand vertex to the given Coxeter diagram the remaining re ection planes pass i 2539coloquio39 2005516 page 33 34 CAP 3 SUBGROUP SEPARABILITY AND SCOTT39S THEOREM through a single nite vertex and these re ections generate the nite Coxeter group E5 Take all the translates of the simplex by this group this yields a polyhedron whose faces all correspond to copies of F Two such copies meet at an angle which is twice the angle of the re ection plane of the hyperbolic simplex which lies between them One sees from the Coxeter diagram that the plane F makes angles 7r2 and 7r4 with the other faces of the hyperbolic simplex7 so the resulting polyhedron is all right as required D We can now construct Ad Lemma 384 Let f be the quadratic form lt 11711711771gt Then for all d S00 Z contains a group Ad which is conjugate to a subgroup of nite index in the Bianchi group PSLQ7 0d The proof requires an additional lemma Assume that j is a diag onal quaternary quadratic form with integer coef cients of signature 31 so that j is equivalent over R to the form lt 11171 gt Let a E Z be a squarefree positive integer and consider the seven dimensional form ja lt a7a7a gt 69 j where 69 denotes orthogonal sum Being more precise7 if we consider the 7dimensional Qvector space V equipped with the form ja there is a natural 4dimensional subspace V0 for which the restriction of the form is j Using this it easily follows that7 Lemma 385 In the notation above the group S00 Z is a sub group of SOja Z D Proof of Lemma 384 Let pd be as above The key claim is that gay lt d7d7d gt 69 pa is equivalent over Q to the form Assuming this claim for the moment7 by Lemma 372 we de duce that there exists Rd E GL77 Q such that RdSOqdZR1 and SO Z are commensurable This together with Lemma 385 gives the required group Gd To prove the claim7 since every positive integer can be written as the sum of four squares7 write d w2 12 y2 22 Let Ad be the 7 X 7 matrix 2539coloquio39 2005516 page 34 SEC 3 s 35 w z y 2 0 0 0 71 w 72 y 0 0 0 7y 2 w 71 0 0 0 72 7y z w 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 Note Ad has determinant d2 so is in GL77 Let F the diagonal matrix associated to the form f and Q0 be the 7 X 7 diagonal matrix of the form lt d7d7d7d7171771gti1e1 of lt d7d7d gt e pd Then a direct check shows that AdFA Q0 as is required D Remark The polyhedron Q is nite covolume since there is only one in nite vertex deleting the plane corresponding to the left hand vertex of the Coxeter group is the only way of obtaining an in nite group and this group is a 5 dimensional Euclidean Coxeter group By inspecting other Coxeter diagrams it can be shown that there are ideal all right polyhedra in Hk at least for 2 S k S 8 This was exploited in 22 to prove that many other discrete subgroups of lsomH are separable on geometrically nite subgroups for 2 S n S 8 1 2539coloquio39 2005516 page 35 Chapter 4 New directions While examination of subgroup separability in the geometric context has generated a good deal of interesting mathematics it is unclear the extent to which it will resolve the central conjectures which it was introduced to resolve For example although it seems somewhat unlikely one might be able to show that hyperbolic 3manifold groups were subgroup separable and this on its own actually answers none of the questions in which we were originally interested In this section we shall formulate some more recent work which appears to be more germane We begin with a simple de nition that underpins much of what follows De nition 406 Let G be a group and H a subgroup Then a homo morphism 6 H A A extends over the nite index subgroup V S G if OHSV 0 There is a homomorphism 9 V a A which is the homomor phism 9 when restricted to H One of the motivations in our setting for making this de nition is the following straightforward theorem Theorem 407 Suppose that G is LERF and H a nitely generated subgroup Suppose t9 H A A is a homomorphism onto a nite group 36 25 39cologuio 392 f 2005515 page 36 37 Then there is subgroup of nite index V of G containing H and a homomorphism 9 V A A with lH 9 Proof We can assume that H has in nite index in G Let K ker 9 a nite index subgroup of H and hence nitely gener ated Since G is LERF there is a nite index subgroup K lt C such that K H K De ne A hK h 1 h E Note that since K is normal in H K lt A and moreover since K has nite index in H the above intersection consists of only a nite number of conjugates of K Hence A has nite index in K and also G Let V denote the group generated by H and A It is easy to check that by construction A is a normal subgroup of V so that V HA The canonical projection 9 V A VA de nes the required extension In other words if the ambient group is LERF then every homomor phism of a nitely generated subgroup onto a nite group can be extended to some subgroup of nite index in G We shall say that the homomorphism t9 virtually extends and that G has the local ex tension property for homomorphisms onto nite groups It turns out that these extension properties give elegant expressions of several well established notions for example residual niteness Theorem 408 G is residually nite if and only ifG has the prop erty that for each of its cyclic groups there is a virtual extension of at least one of the maps onto a nontrivial cyclic group Proof If G has the stated extension property then given a non identity g E G we can extend some homomorphism of ltggt Zh to V Zh Where V has nite index in G Then the kernel of this map has nite index in G and excludes g so that G is residually nite Conversely suppose that G is residually nite and let some nonidentity g E G be given Choose some normal subgroup of nite index in G N say Which excludes g Then the map NW A NvltygtN E lt9gtltykgt for some 16 This extends the map ltggt A Zh this map being non trivial by choice of N i 25 39cologuio 392 2005515 page 37 38 CAP 4 NEW DIRECTIONS Another nice application of Theorem 407 is the following Theorem 409 SLn7 Z is not subgroup separable for all n 2 3 Proof Assume to the contrary that SLn7 Z is subgroup separable Then since SLn7 Z contains free subgroups of all possible ranks7 given an nite group G Theorem 407 constructs a nite index sub group I G of SLn7 Z and an onto homomorphism t9 PG A G However7 SL n7 Z for n 2 3 has the Congruence Subgroup Property ie every subgroup F of nite index in SLn7 Z contains the kernel of some reduction homomorphism SLn7 Z A SLnZmZ7 for m E Z7 m 2 2 Thus kert contains a group of the form Fm for some m 2 2 It follows that G is a quotient of a subgroup of SLnZmZ However7 as we discuss below7 this is impossible for certain groups G For example we can choose G Ag for Z very large7 and we obtain a contradiction To establish the existence of the groups G we use the following lemma see for example 20 Window 2 Lemma 4010 Let p be a prime let SLn7 p denote the nite group SLn7 ZpZ For xed n ing is a quotient of a subgroup ofSLn7 p then n 2 MT F Thus for large enough Z the alternating group Ag is not a quotient of a subgroup of SLnp This lemma completes the proof since by the structure theory of the nite groups SLn7 ZmZ7 given the prime factorization m pmlp p it can be shown essentially the Chinese Remainder Theorem that SLn7 ZmZ E H SLn7 Zpgn Z Furthermore7 the homomorphism SL n7 Zplm Z A SL n7p has ker nel a nite pgroup Putting these statements together it follows that if G Ag as above7 so that in particular Ag is simple7 elementary nite group theory shows that Ag is necessarily a quotient of a sub group of SLnp which is false by Lemma 4010 D i 25 39cologuio 392 2005515 page 38 SEC 41 39 41 An important generalization of the extension property is the follow ing De nition 411 Let G be a group and H a subgroup Then G Virtually retracts to H there is a nite index subgroup V ofG with OHSV 0 There is a homomorphism t9 V a H which is the identity when restricted to H The nite index subgroup V will be called a retractor One should regard a Virtual retraction as an extension of the identity homomor phism H 8 H7 to some nite index subgroup V of G and clearly7 given any such retraction7 we can extend any homomorphism H A A over the nite index subgroup V While retractions are presumably somewhat rare7 one can still ask for their existence in more restricted circumstances7 for example7 we might require that H be a nite sub group7 or an in nite cyclic subgroup or a geometrically nite or quasiconvex subgroup in some more geometric setting The most important incarnation of this comes from the following conjecture Conjecture 412 Suppose that G is the fundamental group of a closed hyperbolic 3manifold Then G virtually retracts to any of its cyclic subgroups A group that satis es this conjecture is said to virtually retract over its cyclic subgroups We note that although this question is signi cantly stronger than the traditional Virtual Betti number conjecture7 phrased in these terms it places the question closer in spirit to questions about extensions of cyclic groups as in the notion of residual niteness We also note that it neither implies7 nor is implied by LERF For example if M is a closed 3manifold that is modeled on the SOL geometry7 then 7r1 is LERF7 but it does not Virtually retract over all of its cyclic subgroups To see this7 since M admits a SOL geometry7 it has a 13 2539coloquio39 2005516 page 39 40 CAP 4 NEW DIRECTIONS nite sheeted cover M1 that is a torus bundle over the circle recall 247 where in this case the surface is a torus7 and it is easy to see that 7r1M1 does not virtually retract onto in nite cyclic groups in the ber group On the other hand7 an examination of the proof of the generalized Scott7s theorem given 4l shows Theorem 413 Suppose that G is a nitely generated group which virtually embeds into an all right hyperbolic Coxeter subgroup of IsomH Then G virtually retracts to its geometrically nite subgroups As with the subgroup separability property7 the property of vir tually retracting over Z is well behaved for subgroups That is to say7 if G virtually retracts over Z and K S G then K virtually retracts over Z This is easily seen If Z lt K is given and V a subgroup of nite index in G which is a retractor for this Z7 then V N K is a retractor for Z in K For nite supergroups there is also a similar result Theorem 414 Suppose that G is a group and K a subgroup of G of nite index hen ifK virtually retracts over Z so does G Proof We begin by noting that from the argument above7 we may assume that K is normal in G Given a ltggt E Z in G choose a retraction r A ltggt K7 where A has nite index in K By the normality of K7 ltggt acts by conjugacy on A while stabilizing ltggt K7 so that we may intersect all these conjugates and restrict the original retraction7 and we may suppose that A is normalised by Choose a faithful linear representation p ltggt O K A GLV7 for some nite dimensional vector space V By composition with the retraction7 we get a nonfaithful linear representation A A GLV Note that AltggtA E N A is finite7 so that we may induce upwards to get a representation p Altggt A GLOW Since A is normal in A ltggt7 the restriction of the induced represen tation p down to A gives Altggt A copies of p which is therefore a faithful representation of ltggt K It follows that when p is restricted to ltggt7 it is faithful on a subgroup of nite index7 namely ltggt N K This forces kerp to be nite and hence trivial7 since Z is torsion free 2539coloquio39 2005516 page 40 SEC 42 41 We have already mentioned the connection with the classical virtual Betti number problem In fact we have more Theorem 415 Suppose that M is a hyperbolic nmanifold for which 7r1M virtually retracts over its cyclic subgroups Then M has in nite virtual Betti number Proof Suppose that the rst Betti number of M is k and let 7 be an element lying in the kernel of the map 7r1 A Let L M A M be a nite sheeted covering in which the lift of some power of 7 becomes an element of in nite order in H1 By considering the transfer map7 we see that with rational coef cients H1 2 H1 EB kerq and the element of lies in kerq It follows that H1 has rank at least k 1 D Remark Thus virtually retracting over cyclic subgroups proves in nite Betti number in a way which seems more natural than the tra ditional approach of nding a surjection to a nonabelian free group It is also rather easy to show that this condition is somewhat more robust than subgroup separability For example7 if A and B virtually retract over their cyclic subgroups7 so does A X B 42 We now discuss two classes of group where virtually retracting over cyclic subgroups can be established without using the full power of LERF Case 1 The Bianchi groups Although the Bianchi groups are known to be LERF by Theorem 3817 and the method of proof shows that the Bianchi groups will virtually retract to all geometrically nite subgroups7 we can give a proof of virtual retraction to in nite cyclic subgroups Because our main interests are in the topology of 3manifolds7 we will work with torsionfree subgroups of nite index in the Bianchi groups to avoid some technicalities Theorem 421 Let 1quot lt PSL20d be a torsionfree subgroup of nite index and 7 E 1quot be a nontrivial element Then 1quot virtually retracts onto lt 7 gt i 25 39coloquio 392 2005515 page 41 42 CAP 4 NEW DIRECTIONS Proof We will assume that 7 is hyperbolic the case of parabolic is similar Let A7 denote the axis of 7 recall 22 The theorem will follow from the next claim Claim There exists a hyperbolic plane H C H3 such that A7 N H in one point FH stabH F acts with nite covolume on H Assuming the claim we proceed to complete the proof By Lemma 323 in the nite volume setting7 FH is separable in 1quot Further more by passage to a subgroup of index 2 if needed we can assume that there is a F1 lt T of nite index7 and 201 Hr1 n rm lt4 H3F1 embeds as a nonseparating orientable surface recall Theorem 322 By assumption A7 N H and so this implies that the projection of A7 to M1 HSFl meets 2H Now the geometric version of separability can be used to nd a further nite sheeted covering for which intersection pairing with 2H de nes a retraction on some power of 7 The proof is completed by the next lemma Lemma 422 Suppose that G is a group which virtually retracts over WU Then G virtually retracts over 7 Proof Suppose that 7r K A Z is a retraction over WW where lt7Tgt lt K and K has nite index in G The element 7 acts by conjugation on K7 stabilising WT so re placing K by all its 7 conjugates7 we may assume that K is normalised by 7 Set K lt K77 gt K lt Then K has nite index in G Moreover7 KK E WW so that K4r K lt 00 Choose some one dimensional faithful representation p Z A C and induce the composition pow up to K to obtain a representation p K A V for some complex vector space V Since K is normal in K7 the restriction of p down to K gives a direct sum of K4r K1 copies of the original representation7 in 2539coloquio39 2005516 page 42 SEC 4 2 43 particular p is faithful on WW and hence faithful on Restricting p to p1p gives the required retractioni D The proof of the claim is completed as follows Firstly part ii of the claim follows from the discussion in 4i5i To prove part of the claim it is easy to see using the density of in C that we can construct a circle C that encloses one of the xed points 6 of 7 and excludes the other and is centered at 20 uovo u0v0 6 00 with radius a small rational number qr Such a circle has an equation of the form l2 7 LEE 42 Expanding clearing denominators and rearranging puts this equation in the form of Lemma 3i5ili D Case 2 Coxeter groups Scott s theorem and the proof of Theorem 381 highlights the importance of groups generated by re ections in the faces of all right polyhedral We now discuss Coxeter groups more generally in the context of virtual retractions to in nite cyclic groups We rst recall some basic statements about Coxeter groups see 16 for details Suppose that W is a group and S is a set of generators all of order 2 Then S is a Coxeter system if W admits a presentation lt S stmlt5gt gt 1gt where mst is the order of s t and there is one relation for each pair st with mst lt 00 We refer to W as a Coxeter group The Coxeter diagram of this presentation consists of a vertex for each element of 5 together with an edge connecting distinct vertices st whenever ms t f 2 and the edge is labelled by msti It is also standard practice in the case when ms t 3 to leave the edge unlabelled and we follow that convention here Since the generators have order 2 this means that if two vertices are not connected by an edge then the generators corresponding to the vertices commuter Thus if the diagram is not connected the Coxeter group is the direct sum of the subgroups given by the connected components A Coxeter group W S is called reducible if its diagram is not connected Otherwise the Coxeter group is irreducible in our context we may as well restrict to irreducible Coxeter groups We shall sketch a proof of see below for de nitions 13 25 39coloquio 392 2005515 page 45 44 CAP 4 NEW DIRECTIONS Theorem 423 Suppose that W is a Coxeter group with all its two generator special subgroups nite Let 7 E W be an element acting hyperbolically on the Coxeter complex Then W virtually retracts over lt 7 gt This implies Corollary 424 A Coxeter group is either virtually abelian or has in nite virtual Betti number Proof If a Coxeter group isnlt virtually abelian7 we can add relations of the form sltk l to nd an in nite nonvirtually abelian Coxeter group with all two generator special subgroups nitel We now form the Davis version of the Coxeter complex We brie y recall the construction Firstly by a special subgroup of W we mean a subgroup lt Squot gt of W where Squot C Squot The nite special subgroups form a poset under inclusion and the Davis complex 2 consists of left cosets of all nite special subgroups where inclusion of faces is de ned by reverse inclusion of cosets ln particular7 if n 5 the n 7 lsimplices correspond to the elements of W and the dual 1 skeleton of E is a modi ed Cayley graph of W with generating set S The modi cation consists of identifying the edge labelled s with the edge labelled 3 1 for each generator 8 E S The action of W on the left cosets by left multiplication induces a simplicial action of W on 2 A top dimensional simplex7 C7 of E is called a chamber Observe that the only element of W which maps some chamber to itself is the identity see 107 Chapter lll7 4 Lemma 6 In his thesis7 it was shown by Moussong that the cells of this complex can be metrized as Euclidean polyhedra so that in the induced piecewise Euclidean metric7 E is a CAT0 space Following 107 Chapter lll7 47 we see that given any pair of adjacent chambers C an 7 t ere is a unique automorphism s of the Coxeter complex of order 2 which exchanges C and C while xing C N 07 and this gives rise to a wall denoted by H5 in the Coxeter complex7 namely H5 Fir Conversely7 any re ection in W gives rise to a unique walll Note that walls separate the Coxeter complex See 107 for example7 Chapter lll 3 Corollary 3 and are totally geodesic in the CAT0 metric7 since they are the xed set of an orientation reversing isometryl i 2539coloquio39 2005516 page 44 SEC 4 2 45 Fix the following notation let 9 6 W7 then CW 9 denotes the centralizer of g in Lemma 425 stabH5 CWs Proof Let 7 be an element of stabH5 Then 7C N C is some codimension l face in H5 and is therefore xed by s It follows that s and 7 1 s 7 are both automorphisms of order two xing C N C pointwise and exchanging 7C with 76 Thus 7 1 s 7 s 1 maps C to itself and therefore is the identity element of W See 107 Chapter 1117 4 Lemma 6 Conversely7 if 7 E JVs then it follows that sWHS 7H5 As s xes a unique wall7 we deduce that 7H5 HS7 and 7 E stabH5 as was required D The key use of this lemma is the following result of 19 For the convenience of the reader we include a proo Theorem 426 19 Let a G A G be an automorphism of a residually nite group G Then Fiza is separable in G Proof Choose an element 7 not lying in Fiza This means that the element 7 1 17 is not the identity element7 so that there is a homomorphism 45 G A F onto a nite group F7 so that 457 1 17 is not the identity element De ne a homomorphism zGeFxF by g 4159 q5a Note that 39i39 maps Fix a into the diagonal subgroup of F X F7 however by construction7 7 77 a7 does not lie in the diagonal subgroup7 so that 1 f7 l f E F is the required subgroup of nite index D It is a theorem of Tits that Coxeter groups are linear7 it follows that they are residually nite and we de u Corollary 427 In the notation above stabHS is a separable sub group of W Fix some element 9 E W which which acts hyperbolically See 9 p 229 in particular7 there is a geodesic line 7 in 2 along which 9 i 25 39coloquio 392 2005515 page 45 46 CAP 4 NEW DIRECTIONS acts as translation As in 9 Theorem 687 the translation distance along 7 is the minimal distance points in E are moved The key claim is now Lemma 428 Then there is a wall X5 so that one end of 7 lies on one side of X5 the other end lies on the other Proof Choose some axis 7 for the element 9 this must meet some wall X5 transversely The wall is totally geodesic so 7 cannot meet it more than once hence the ends of 7 lie on either side of X5 Of course7 this implies in particular that X5 y is nonempty We can use this result to prove the main result Consider the subgroup W7 of index 2 in W which is the kernel of the map W A Z2 given by sending each generator in S to 1 E ZZ the action of W4r on walls is now orientation preserving Moreover7 W is linear so there is a torsion free subgroup of nite index inside W4r which we denote by WT Let stabX5 be the stabilizer of X5 inside the group WT Choose some point p so far towards the 700 end of 7 that the 700 end of 7 never returns to X5 Now choose some very large power gt 6 WT so that g p 10 lies a similarly long way towards the 00 end of 7 ln particular7 p and 10 are on either side of X5 Consider 7r X XstabX5 this contains the compact subcomplex formed by XsstabX5 together with the image of the subarc of 7 between p and 10 Denote this subcomplex by C Since stabX5 acts by isometries which do not exchange the sides7 the points 7rp and 7rp lie on opposite sides of XsstabX5 Moreover7 the ends of the projection of the geodesic 7 never return to XsstabX5 past the points 7rp and 7rp In particular the arc meets XsstabX5 an odd number of times The subgroup stabX5 is separable inside WT so by REF7 there is a subgroup stab XS S K of nite index in WT7 so that the projection of C in the covering XstabX5 XK is an embedding of C Since C is embedded in XK7 and the ends of 7 in the covering XstabX5 never return to XsstabX5 past the chosen points7 so that the lift ofg C XK running through the arc portion of C must meet the surface portion XsstabX5 in an odd number of points It follows that taking intersection number with XsstabX5 gives 2539coloquio39 2005516 page 46 2539Coloquio39 2005516 page 47 3 SEC 42 47 the relevant element of H1XK Z Which retracts some power of g The theorem follows from Theorem 422 Bibliography 1 1 Agol Tameness of hyperbolic 3manifolds preprint 2 1 Agol D D Long and A W Reid The Bianchi groups are separable on geometrically nite subgroups Ann of Math 153 2001 5997621 E G Baumslag and D Solitar Some twogenerator onerelator nonHop an groups Bulll A M S 68 1962 1992011 E R Benedetti and C Petronio Lectures on Hyperbolic Geome try Universitext SpringerVerlag 1992 A Blass and P M Neumann An application of universal algebra in group theory Michigan J Math 21 1974 16771691 E F Bonahon Bouts des vari t s hyperboliques de dimension 5 Annals of Math 124 1986 717158 E A Borel Compact Cli ord Klein forms of symmetric spaces Topology 2 1963 111 7122 E 48 13 2539Coloquio39 2005516 page 48 BIBLIOGRAPHY 49 8 A Borel and Harish Chandra Arithmetic subgroups of alge braic groups Annals of Math 75 1962 4857535 9 M Bridson amp A Hae iger Metric spaces of nonpositive curvature Grundlehren der Mathematischen Wissenschaften 319 SpringerVerlag 1999 10 K Brown Buildings SpringerVerlag 1989 11 R G Burns D Karrass and D Solitar A note on subgroups with separable nitely generated subgroups Bull Australian Math Soc 36 1987 1537160 12 D Calegari and D Gabai Shrinkwrapping and the taming of hyperbolic 3manifolds preprint 13 D Cooper and D D Long Some surface subgroups survive surgery Geometry and Topology 5 2001 3477367 14 M Hall Jr Coset representations in free groups Trans A M S 67 1949 4217432 15 J Hempel Residual niteness for 3Manifolds 1n Combinato rial Group Theory and Topology Ann of Math Studies 111 379 7396 P U P 1987 16 J E Humphreys Re ection Groups and Coxeter Groups Cambridge Studies in Advanced Mathematics 29 CUP 1990 17 T Y Larn The Algebraic Theory of Quadratic Forms Ben jamin 1973 13 2539Coloquio39 2005516 page 49 50 18 l2 7 BIBLIOGRAPHY Dl Dl Long Immersions and embeddings of totally geodesic surfaces Bull London Math Soc 19 1987 4817484 11 Long and Go Niblo Subgroup separability and 3manifold groups Mathl Zeitl 207 1991 209 7 215 Al Lubotzky and Dl Segal Subgroup Growth Progress in Math 212 Birkhauser 2003 C F Miller 111 On grouptheoretic decision problems Annals of Math Study 68 PlUlP 1971 Dl Dl Long and Al Wl Reid On subgroup separability in hyperbolic Coxeter groups Geoml Dedicata 87 2001 2457260 T Li Immersed essential surfaces in hyperbolic 3manifolds Comml Anal and Geoml 10 2002 2757290 Cl Maclachlan and Al Wl Reid The Arithmetic of Hyperbolic SManifolds Gradl Texts in Math 219 SpringerVerlag 2003 1 Go Ratcliffe Foundations of Hyperbolic Manifolds GlTlMl 149 SpringerVerlag 1994 Go P Scott and C T C Wall Topological methods in group theory in Homological Group Theory LlMlSl Lecture Notes 36 1377203 Cambridge University Press 1979 Gl P Scott Subgroups of surface groups are almost geometric 1 London Math Soc 17 1978 555 565 See also ibid Correction 1 London Math Soc 32 1985 2177220 13 2539coloquio39 2005516 page 50

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