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# Topics In Math MAT 280

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This 29 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 280 at University of California - Davis taught by Michael Kapovich in Fall. Since its upload, it has received 6 views. For similar materials see /class/187390/mat-280-university-of-california-davis in Mathematics (M) at University of California - Davis.

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PROBLEMS ON BOUNDARIES OF GROUPS AND KLEINIAN GROUPS MISHA KAPOVICH Most problems in this list were collected during the workshop Boundaries in Geometric Group Theory 7 in AlM7 2005 1 BACKGROUND Ideal boundaries of hyperbolic spaces Suppose that X is a hyperbolic metric space Pick a base point 0 E X This de nes the 070mm product zy0 E R for points x y E X The ideal boundary dooX of X is the collection of equivalence classes of sequences in X where if and only if lim 77y70 oo Hoe The topology on dooX is de ned as follows Let g E 600X De ne r neighborhood of g to be U r 77 E dooX 3a with g xiL77 lignjorifiyj0 2 7 Then the basis of topology at g consists of U 7 7quot 2 0 We will refer to the resulting ideal boundary 600X as the Gromoviboundary of X One can check that the topology on dooX is independent of the choice of the base point Moreover7 if f X a Y is a quasi isometry then it induces a homeomorphism 3007 600X a 3005 The Gromov product extends to a continuous function 677 i aooX X aooX 0700l The geodesic boundary of X admits a family of visual metrics dgo de ned as follows Pick a positive parameter 1 Given points 777 6 dooX consider various chains 0 17 m where m varies so that 1 7 m 77 Given such a chain7 de ne mil dc 777 Z 5UEi i1o7 H 7 where 6quot 0 Finally7 dioww irclf 14677 where the in mum is taken over all chains connecting g and 77 Taking different values of 1 results in Ho39lderiequivalent metrics Each quasi isometry X a Y yields a quasi symmetric homeomorphism see section 8 for the de nition 600X7dgo a 600K Conversely7 each quasi symmetric homeomorphism as above extends to a quasi iso metry X a Y7 see 51 Date October 24 2007 2 MlSHA KAPOVICH Then the ideal boundary of a Gromovihyperbolic group G is de ned as aooPGv where PC is a Cayley graph of G Hence PC is well de ned up to a quasi symmetric homeomorphism Ideal boundaries of GAT0 spaces Consider a GAT0 space X Two geodesic rays oz R a X are said to be equivalent if there exists a constant C E R such that dat t S GVt E Kr The geodesic boundary dooX of X is de ned to be the set of equivalence classes oz of geodesic rays oz in X Fix a base point o E X If X is locally compact which we will assume from now on7 then there exists a unique a representative oz in each equivalence class oz so that oz0 0 With this convention the visual topology on dooX is de ned as the compact open topology on the space of maps R a X One can check that this topology is independent of the choice of the base point and that isometries X a Y induce homeomorphisms dooX a 3005 Example If X R then dooX is homeomorphic to Sn l If X is a GAT71 space then it is also Gromov hyperbolic Then the two ideal boundaries of X one de ned via sequences and the other de ned via geodesic rays are canonically homeomorphic to each other More speci cally7 each geodesic ray oz de nes sequences oztl7 for ti 6 R diverging to in nity The equivalence class of such is independent of and one gets a homeomorphism from the GAT0 boundary to the Gromov boundary ln general7 quasi isometries of GAT0 spaces to not extend to the ideal boundaries in any sense Moreover7 Bruce Kleiner and Chris Croke constructed examples 22 of pairs of GAT0 spaces XX which admit geometric ie isometric7 discrete7 cocompact actions by the same group G so that 300X7 dooX are not homeomorphic Therefore7 given a GAT07group G one can talk only of the collection of GAT0 boundaries of G7 ie the set dooX 3G m X where the actions G n X are geometric 2 TOPOLOGY OF BOUNDARIES OF HYPERBOLIC GROUPS Problem 1 Misha Kapovich What spaces can arise as boundaries of hyperbolic groups As a sub problem For which k do k dimensional stable Menger spaces appear as boundaries Example 1 Damian Osajda Let X be a thick right angled hyperbolic building of rank n 17 ie with apartments isometric to EFL Then the ideal boundary of X is a stable Menger space Mmk However n 1 dimensional right angled hyperbolic re ection groups exist only for n S 3 Problem 2 Can one remove the right angled77 assumption in Osajda result Background The Menger space Mk is obtained by iteratively subdividing an n cube into 3 subcubes and removing those that do not touch the k skeleton7 see 4 PROBLEM LIST 3 for a detailed discussion of the topology of these spaces Below are few properties of Mk 0 Mk has topological dimension k 0 Mk is stable when n 2 2k 1 that is replacing n by a larger value does not change 0 Any k dimensional compact metric space embeds in some stable Mk Problem 3 Panos Papasoglu What 2 dimensional spaces arise as boundaries of hyperbolic groups Can restrict to cases with no virtual splitting no local cut points or cut arcs and no Cantor set that separates Background 2 dimensional Pontryagin surfaces and 2 dimensional Menger spa ces M15 appear as boundaries of hyperbolic Coxeter groups see 27 According to work of Misha Kapovich and Bruce Kleiner 37 if 300G is 1 dimensional connected and has no local cut points then 300G is homeomorphic to a Sierpinski carpet M12 or the Menger space M13 Problem 4 Mike Davis Are there torsion free hyperbolic groups G with chGchG lt 23 7 Background Here cdR is the cohomological dimension over a ring R Mladen Bestvina and Geoff Mess 6 have shown that a For torsion free hyperbolic groups cdRG cdR600G 1 b There are hyperbolic groups G such that chG 3 and chG 2 Problem 5 Nadia Benakli What can be said about boundaries arising from strict hyperbolizatz39on constructions of Charney and Davis 18 Problem 6 llia Kapovich ls there an example of a group G which is hyperbolic relative to some parabolic subgroups that are nilpotent of class 2 3 whose Bowditch boundary is homeomorphic to some n sphere Remark 1 Tadeusz Januszkiewicz Strict hyperbolization of piecewise linear mani folds gives many examples of hyperbolic groups G with 300G homeomorphic to S Problem 7 Misha Kapovich Suppose that Z is a compact metrizable topological space G n Z is a convergence action which is topologically transitive ie each G7 orbit is dense in Z ls there a Gromov hyperbolic space X with the ideal boundary Z so that the action G n Z extends to a uniformly quasi isometric quasi action G n X Background Suppose that Z is a topological space Z3 is the set of triples of distinct points in Z The space Z3 has a natural topology induced from Z3 A topological group action G n Z is called a convergence action if the induced action G n Z3 is properly discontinuous A convergence action G n Z is called uniform if Z3G is compact Examples of convergence group actions are given by uniformly quasi Moebius actions G n Z eg are induced on Z dooX by uniformly quasi isometric quasi actions G n X Brian Bowditch 12 proved that each uniform convergence action G n Z is equivalent to the action of a hyperbolic group on its ideal boundary 4 MISHA KAPOVICH Problem 8 Tadeusz Januszkiewicz Find topological restrictions on the ideal bound aries of GAT71 cubical complexes Background A GAT71 cubical complex is a GAT71 complex X where every n cell is a combinatorial cube isometric to a polytope in 1111 so that the isom etry preserves the combinatorial structure For instance such a complex can cover closed hyperbolic 3 manifold It was proven by Januszkiewicz and Swiatkowski 35 that ooX cannot be homeomorphic to S4 Moreover 600X cannot contain an essen tial k sphere for k 2 4 3 BOUNDARIES OF COXETER GROUPS Let G be a nitely generated Coxeter group with Coxeter presentation SRgt This presentation determines a Davis Vmberg sampler X see 24 whose dimension equals rank of the maximal nite special subgroup of G with respect to the above presentation The complex X admits a natural piecewise Euclidean GAT0 metric The group G acts on X properly discontinuously and cocompactly Hence X has visual boundary 3X which we can regard as a boundary of G Topology of 3X was studied in 27 28 For instance 27 constructs examples of hyperbolic Coxeter groups whose boundaries are both orientable and non orientable Pontryagin surfaces and 2 dimensional Menger compacta Recall that a Pontryagin surface is obtained as follows Let K be a connected compact without boundary triangulated surface De ne PK by replacing each closed 2 simplex 039 in K with a copy Ka of the closure of K 039 We get the map PK a K by sending each K0 to 039 Set Pn PPn1 Then the corresponding Pontryagin surface Po0 based on P0 is inverse limit of the sequence UP a PnA a a P1 a P0 It turns out that Po0 can have only three distinct topological types 1 If P0 SZ then POO g SZ 2 If P0 is oriented but has genus 2 1 then Po0 is oriented ie H2POOZ Z but not homeomorphic to 82 3 If P0 is not oriented then Po0 is unoriented In this case the rational homological dimension of Po0 equals 1 Problem 9 Alexander Dranishnikov Is it true that isomorphic Coxeter groups have homeomorphic boundaries Remark 2 It appears that the answer is positive provided that all labels are powers of 2 REFERENCE Problem 10 Alexander Dranishnikov Does there exist a Coxeter group Gn with n dimensional boundary 6G so that the rational homological dimension of 3G equals 1 Problem 11 Alexander Dranishnikov Under which conditions on the Coxeter dia gram of G the boundary of a Coxeter group is n connected and locally n connected PROBLEM LIST 5 Partial results in this direction are obtained in 26 The main motivation for this problem comes from the problem of realizing Menger spaces as boundaries of Coxeter groups Problem 12 Misha Kapovich Can exotic homology manifolds as in 14 appear as ideal boundaries of Coxeter groups 4 UNIVERSALITY PHENOMENA The term universality loosely describes the following situation There is a class C of groups spaces of different nature whose ideal boundaries are all homeomorphic Usually such results come from topological rigidity results for certain families of compacta Examples of universality phenomena 1 Consider the class of all 2 dimensional hyperbolic groups which are 1 ended do not split over virtually cyclic groups are not commensurable to surface groups are not relative PD3 groups Then the ideal boundaries of all groups in this class are homeomorphic to the Menger curve See 37 2 The boundaries of the right angled rank n1 hyperbolic buildings in Example 1 are all homeomorphic since they are all homeomorphic to the stable Menger space 3 Let N be a closed n manifold A be its triangulation Then A determines a right angled Coxeter graph 00N A and 71 1 dimensional David Vinberg complex CN A We assume in addition that A is a ag complex satisfying the no square condition which guarantees hyperbolicity of the resulting Coxeter group Suppose A1 A2 are two such triangulations of N which admit a common subdi vision Let Cl CNA Then Fischer 32 60001 Note that 6000N A 6000NNAA In particular the boundaries which appear in case n 2 are of three types 82 oriented Pontryagin surface non orientable Pontryagin surface Problem 13 TadeusZ Januszkiewicz Find more universality phenomena Problem 14 Misha Kapovich ls it true that 6000N A is a topological invariant of N7 Problem 15 Misha Kapovich Tadeusz Januszkiewicz Suppose that N1 A1 and N2 A2 are closed 3 manifolds equipped with ag triangulations so that 6000N1 A1 6000N2 Does it follow that every prime connected sum summand of N appears as a connected sum summand of Ni i 1 2 What can be said in higher dimensions 6 MISHA KAPOVICH 5 MARKOV COMPACTA The notion of Markov compactum is a generalization of the boundary of a group Let IC Kl7 gt 17i 2 0 be an inverse system of nite simplicial complexes 39 1 452 IQ1 a For a simplex o E K let 1o denote the inverse subsystem K0 formed by the subcomplexes building blocks Ki1c7 3 171Ki07j Z Z Km 3 U The inverse system IC is called Markov if it contains only nitely many isomor phism classes of inverse subsystems K0 A Markov compactum is a compactum ob tained as the inverse limit of a Markov inverse system Thus IC is obtained from K0 by inductively replacing simplices o in Kl with the building blocks KHLU using only nitely many replacement rules For instance7 the Pontryagin surfaces are Markov compacta Markov compacta appear naturally as boundaries of hyperbolic and Coxeter groups For every compactum Z either dim Z n dimZ or dim Z n 7 1 dimZ 1 In the latter case7 Z is called a Boltyansky compactum Problem 16 Alexander Dranishnikov Let Z be a compactum which is a Z boundary of a group G Then Z is never a Boltyansky compactum In the special case when Z is an Markov compactum7 so that all building blocks K0 a o are isomorphic7 it was proven in 29 that Z cannot be a Boltyansky compactum 6 BOUNDARIES OF OAT0 SPACES Problem 17 Kim Ruane Examples of Kleiner and Croke 227 23 of non unique boundaries are badly non locally connected ls that essential in having the exibil ity77 to have many boundaries That is7 does local connectedness imply uniqueness of the boundary in the 1 ended case for CAT0 groups Background Suppose that XY are Gromov hyperbolic spaces and f X a Y is a quasi isometry Then f extends naturally to a homeomorphism doof 600X a 3005 In particular7 the ideal boundaries of X and Y are not homeomorphic The situation for the GAT0 spaces is quite different De nition 1 A group action G n X on a metric space X is called geometric if it is isometric7 properly discontinuous and cocompact For a CAT0 group G acting geometrically on spaces Xi there is an induced action of G on the boundary dooXi For G spaces X1 and X27 the boundaries may be a non homeomorphic7 or b homeomorphic7 but not G equivariantly PROBLEM LIST 7 The Croke Kleiner examples are torus complexes which are combinatorially the same but where the angle 04 between the principal circles varies 23 22 showed that these complexes K0 which all have the same fundamental group a right angled Artin group in particular have universal covers whose boundaries are not homeomorphic when 04 7T2 and Oz 31 7T2 Julia Wilson showed that any two distinct values of 04 give non homeomorphic boundaries Problem 18 Dani Wise Suppose that G is a GAT0 group which does not split over a small subgroup Does it follow that 300G is unique Problem 19 Dani Wise ls the boundary well de ned for groups acting geomet rically on GAT0 cube complexes More precisely suppose that X1X2 are cube complexes which admit geometric actions of a group G Does it follow that 600X1 630ng Problem 20 Ross Geoghegan What topological invariants distinguish boundaries In particular what topological properties of boundaries are quasi isometry invariants Does something coarser than the topology stay invariant Remark 3 All boundaries for a given group are shape equivalent so cannot be dis tinguished by their Cech cohomology See 30 for the de nition of shape equivalence It was shown by Eric Swenson 57 that for a proper cocompact GAT0 space X the ideal boundary dooX has nite topological dimension It was shown by Ross Geoghegan and Pedro Ontaneda 33 that the topological dimension of 600X is a quasi isometry invariant of X Here and below a space X is called cocompact if IsomX acts cocompactly on A useful class of maps is called cell like inverse images of points are compact metrizable and each is shape equivalent to a point For a nite dimensional compact subset Y of R or of any ANR shape equivalent to a point77 is equivalent to saying Y can be contracted to a point in any of its neighborhoods7 Remark 4 Cell like maps are simple homotopy equivalences Problem 21 Ross Geoghegan If G acts geometrically on two CAT0 spaces are the resulting boundaries cell like equivalent That is does there exist a space Z with cell like maps to each of the two spaces Remark 5 Ric Ancel Craig Guilbault and Julia Wilson have some examples when the answer is positive they showed that the complexes K0 see Croke Kleiner exam ples above are all cell like equivalent Suppose that G m X 239 12 are isometric cocompact properly discontinuous actions of G on two CAT0 spaces Problem 22 Thomas Delzant ls there a convex core for the diagonal action of G on X1 gtlt X2 A special case is surface groups G with X1 and X2 corresponding to different hyperbolic structures If there is a convex core can Z the space with cell like maps to X1 and X2 be taken to be the boundary of the core Remark 6 Bruce Kleiner Convex sets are actually rare see Remark 31 so maybe there is a different problem with better prospects 8 MISHA KAPOVICH Danny Calegari One can try to de ne a new ideal boundary for GAT0 spaces which is different from the visual boundary dooX by looking at the space of all quasi geodesics in X For example in R2 consider all equivalence classes of K quasi geodesics with the compact open topology Varying K gives a ltration of the space of all quasi geodesics Can one do interesting analysis on such a space Problem 23 Danny Calegari De ne a topology on the set of quasi geodesics in a proper geodesic or coarsely homogeneous or cocompact GAT0 space which 1 has a description as an increasing union of compact metrizable spaces 2 has an inclusion of its visual boundary ooX into it 3 is quasi isometry invariant 4 has reasonable measure classes which are quasipreserved According to a theorem by Brian Bowditch and Gadde Swarup 13 56 if G is a 1 ended hyperbolic group then 300G has no cut points For G a CAT0 group a theorem of Eric Swenson says that if c E 300G is a cut point then there is an in nite torsion subgroup of G xing 0 Problem 24 Conjecture Eric Swenson Any GAT0 group has no in nite torsion subgroups A Euclidean retract is a compact space that embeds into some R as a retract A compact metrizable space Z is a Z set in X if it is homotopically negligible77 for every open U C X the inclusion UZ in U is a homotopy equivalence A Z strueture on a group G is a pair XZ such that o X is a Euclidean retract 0 Z is a Z set in X o X X Z admits a covering space action of G with XG compact o the set of translates of any compact set K C X is a null sequence in X that is for each 6 gt 0 there are only nitely many translates with diam gt 6 Finally Z is a boundary of G or Z structure boundary if there exists a Z structure X Z on G The above notion boundary of G was generalized by T Farrell and J Lafont as follows Ar EZ bourzdary of a group G is a boundary Z QEZG so that the action of G on X extends to topological action of G on Z Problem 25 Misha Kapovich Let G be a hyperbolic group and QEZG be its EZ boundary ls it true that QEZG is equivariantly homeomorphic to the Gromov boundary of G Problem 26 Mladen Bestvina Can there be two different boundaries in the sense of Z structures for a group G that are not cell like equivalent Remark 7 Note that this problem is even open for Z For CAT0 spaces the visual boundaries are Z structure boundaries so Problem 21 is a special case Problem 27 Bruce Kleiner ls the property of splitting over a 2 ended subgroup an invariant of Bestvina boundaries PROBLEM LIST 9 Some necessary conditions are known for compact7 metrizable spaces X to be the boundary of some proper cocompact GAT0 space 1 X should have 127 or in nitely many components 2 X is nite dimensional Theorem of Swenson 3 X has nontrivial top Cech cohomology Geoghegan Ontaneda In the case when X admits a cocompact free action by a discrete subgroup of isometries7 one necessary condition is due to Bestvina the dimension of every nonempty open set U C X is equal to the dimension of X Problem 28 Ross Geoghegan Extend these lists7 or give a complete classi cation Problem 29 Kevin Whyte Does every CAT0 group have nite asymptotic di mension 7 ASYMPTOTIC TOPOLOGY Problems below are mostly motivated by the following rigidity results of Panos Papasoglu7 50 Theorem 1 Suppose that G is a nitely presented 1 ended group Then 1 The JSJ decomposition of G is invariant under quasi isometries 2 A quasiline coarsely separates Cayley graph of G iff G splits over virtually Z or G is virtually a surface group 3 No quasi ray coarsely separates the Cayley graph of G Problem 30 Panos Papasoglu Do these results hold for general nitely generated groups Problem 31 Panos Papasoglu Are splittings over Z2 or Z invariant under quasi isometry The analogous problem also makes sense for the JSJ decompositions Problem 32 Panos Papasoglu Suppose G is nitely generated and there is a sequence of quasicircles that separate its Cayley graph ls G virtually a surface group Problem 33 Conjecture of Panos Papasoglu If G is nitely generated with as ymptotic dimension 2 n7 and X is a subset of the Cayley graph with asymptotic dimension 3 n 7 2 that coarsely separates the Cayley graph7 then G splits over some subgroup H S G with asymptotic dimension 3 n 7 1 A homogeneous continuum is a locally connected compact metric space whose group of homeomorphisms acts transitively Papasoglu showed that every simply connected homogeneous continuum has the property that no simple arc separates it Problem 34 Panos Papasoglu Do all homogeneous continua with dimension greater than 2 have this property 8 ANALYTICAL ASPECTS OF BOUNDARIES OF GROUPS We begin with the basic de nitions of the quasiconformal analysis For a quadru ple of points 711240 in a metric space X7 mg7211 denotes their cross ratio7 ie my 6127 w lylzlw dyz dwx39 Note MISHA KAPOVICH Quasieoaformal analytic de nition A homeomorphism f R 7 R is quasieoaformal iff 1 f e Wli nn and 2 There exists K Kfz lt 00 so that is the Jacobian of f The essential supremum Kf of Kfz on R is called the coef cient of qua siconformality of f A mapping f is called K quasieoaformal if Kf S K W S KJfx ae here Jf The map f is differentiable almost everywhere7 so the derivative and Jacobian in 2 make sense pointwise ae The assumption 1 can be replaced by the assump tion that f is ACL7 ie7 that f is absolutely continuous on ae line parallel to the n coordinate directions The above analytical de nition of quasiconformality for maps of R turns out to be equivalent to four other de nitions given below 1 A D V Quasieoaformal metric de nition Let f R 7 R be a homeomorphism For 7 E R4 de ne the following LAW suplfy 7 WM i la 7 zl 7 7 MWquot 31nflfy 7 WM i la 7 gel 7 7 Hfx lim sup wal T THO f 7 Then7 f is quasieoaformal iff Hfz is uniformly bounded by some H Hf 2 1 Quasieoaformal geometric de nition Let P be a family of paths in R We say a Borel function p R a 07 00 is admissible for P iff for every 7 E P we have fyp d5 21 De ne modnP inf fw p z dx p is admissible for P A homeomor phism f R 7 R is quasieszormal iff there is some constant K Kf 2 1 such that for every path family P we have the metric dilatation of f modnl modnfl Kmodnl A homeomorphism f X 7 Y between metric spaces X and Y is quasisym metric iff there exists a homeomorphism i7 07 oo 7 07 00 such that for all triples of distinct points Luz 6 X7 the following inequality holds lfW 7 MIN S n lit 7 lf957f2l l9672l A homeomorphism f X 7 Y between metric spaces X and Y is quasi Moebius iff there exists a homeomorphism i7 000 7 000 such that for all quadruples of distinct points 711240 6 X7 the following inequality holds lf7fy7f27fwl S Whit727M Note that De nitions 174 make sense in the context of general metric spaces7 see below for details If X is noncompact then quasi symmetric maps are the same as quasi Moebius maps However7 for compact metric spaces quasi Moebius is a more appropriate although more cumbersome de nition One can rectify this problem by rede ning quasi symmetric maps for compact metric spaces as follows A map PROBLEM LIST 11 f X a Y is quasi symmetric if X admits a nite covering by open spaces Ul C X so that the restriction flUl is quasi symmetric in the above sense for each i One de nes a quasi symmetric equivalence for metric spaces by X qu Y if there exists a quasisymmetric homeomorphism X a Y Let X7d7p be a metric measure space7 where a is a Borel measure Then X is called Ahlfors Q regular7 if there exists 0 2 1 so that O lRQ BRx ORQ for each R S diamX Remark 8 Let X be a metric space with the Hausdorff dimension HdimX Q Then the most natural measure to use is the Q Hausdorff measure on X This is the measure to be used for the boundaries of hyperbolic groups Then X7d is called Ahlfors regular if it is Ahlfors Q regular with Q HdimX Given two compact continua E7 F in a metric space X de ne their relative distance dEF mindiamE7 diamF39 Here dEF mindzy z E Ey E Given an Ahlfors Q regular metric measure space X7 de ne monEF to be monP where P is the set of all curves in X connecting E to F AEF De nition 2 A metric space X is called Q Loewrier if it satis es the inequality monEF S 15AEF7 for a certain function 1 Problem 35 Mario Bonk Are diffeomorphisms R a R dense in the space of all quasiconformal maps Remark 9 Juha Heinonen The answer is known to be yes77 for n 23 due to Moise7s theorem Remark 10 Misha Kapovich ln fact7 the answer is also known to be 77yes77 for qua siconformal diffeomorphisms of R n gt 4 This was proven by Connell 20 for stable homeomorphisms of R n 2 7 improved by Bing 7 to cover dimensions 2 5 Lastly7 it was shown by Kirby 39 that all orientation preserving homeomorphisms of R are stable for n 2 5 Note that the proof in the case of quasiconformal homeomorphisms is easier since quasi conformal homeomorphisms are differentiable ae and the stable homeomorphism conjecture was known for n 2 5 prior to Kirby7s work However the problem appears to be open in the case n 4 On the other hand7 Kirby ob served that for suf ciently large ii there are open connected subsets 91 92 C R and a homeomorphism f 91 a 92 which cannot be approximated by diffeomorphisms fj 91 a 92 12 MISHA KAPOVICH The problem becomes more subtle if we require approximating diffeomorphisms to be globally quasiconformal Problem 36 Misha Kapovich Let f B a B be a quasiconformal home omorphism Can f be approximated by globally quasiconformal diffeomorphisms fj B a B Can this be done so that fs are K quasiconformal for all j Note that all the maps in problem will extend quasiconformally to the closed n ball Problem 37 Mario Bonk Find good classes of spaces such that the in nitesimal metric condition for quasiconformality implies the local condition This is generally true in Loewner spaces Problem 38 Kim Ruane Outside of the boundaries of Fuchsian buildings what boundaries have the Loewner property Remark 11 Loewner spaces are good for analytic tools have a cotangent bundle so can do calculus also can do PDEs etc Problem 39 Juha Heinonen Let X be a non smoothable closed simply connected 4 manifold Does it admit an Ahlfors 4 regular linearly locally contractible metric This is wide open unknown even for examples like E8 Remark 12 The non smoothable closed simply connected 4 manifolds like E8 are known not to admit a quasiconformal atlas 25 In dimensions 2 5 Sullivan 55 proved that every topological manifold admits a quasiconformal atlas and moreover quasiconformal structure is unique An alternative proof of Sullivan7s theorem and its generalization was given by J Luukkainen in 42 his proof avoids the construction of almost parallelizable hyperbolic manifolds see also 59 It was observed by Tom Farrell that a detailed proof of the fact that all closed hyperbolic n manifolds are virtually almost parallelizable and much more is contained in the paper by B Okun 48 Hence in dimension n 2 5 one would ask for Ahlfors n regular linearly locally contractible metrics on the unresolvable homology manifolds see 14 The broad goal here is to nd an analytic framework for studying erratic topological and homology manifolds Problem 40 Uri Bader Develop a theory for analysis on the ideal boundaries of relatively hyperbolic groups as it is done for hyperbolic groups Problem 41 Bruce Kleiner In what generality does quasiconformal imply qua sisymmetric Speci cally of interest are self similar spaces which are connected without local cut points or visual boundaries of hyperbolic groups De nition Call a metric space X quasi isometrically cohop arz if each quasi isometric embedding f X a X is a quasi isometry Examples include Poincare duality groups solvable groups Baumslag Solitar groups X is quasisymmet cally cohop arz if every quasisymmetric embedding X a X is onto The above de nition is a coarse analogue of the notion of cohop arz groups from group theory PROBLEM LIST 13 Fact a hyperbolic group G is quasi isometrically cohop an iff 300G is quasisym metrically cohop an7 cf 51 Problem 42 llia Kapovich Take your favorite metric fractal ls it quasisymmet rically cohop an What about the boundaries of hyperbolic groups Subproblem What about the case of round Sierpinski carpets and Menger spaces which appear as boundaries of hyperbolic groups Background Rourid Sierpinski carpets are the ones which are bounded by round circles Such sets arise as the ideal boundaries of fundamental groups of compact hyperbolic manifolds with nonempty totally geodesic boundary It is known that if G76quot are such groups which are not commensurable then their ideal boundaries are not quasisymmetric to each other There is a similar rigidity theorem due to Marc Bourdon and Herve Pajot 10 for a certain class of Menger curves7 ie the ones which appear as visual boundaries of 2 dimensional Fuchsian buildings Quasisymmetric cohop an property is open in both cases Remark 13 Danny Calegari As an example for the previous problem the limit set L of a leaf of a taut foliation of a hyperbolic 3 manifold with 1 sided branching is a dendrite in S2 which is nowhere dense7 has Assouad dimension 27 and for any point p in L and any neighborhood U ofp in 827 L can be embedded by a conformal automorphism of 52 into L U Remark 14 Juha Heinonen If X is the staridard square77 Menger space Mk then it is clearly not quasisymmetrically cohop an Problem 43 Conjecture Juha Heinonen lf 300G is Loewner7 then it is quasisym metrically cohop an Boundaries of Fuchsian buildings provide a good test case for this conjecture Problem 44 Bruce Kleiner If G is a hyperbolic group and 300G is connected with no local cut points7 is there a natural measure class which is quasisymmetrically invariant That is7 invariant under quasisymmetric homeomorphisms 3006 a 600G Motivation rigidity theorems rely on absolute continuity of quasisymmetric maps as a foundational ingredient Remark 15 If 300G is Loewner7 then the answer is yes77 But there are examples of Bourdon and Pajot 11 whose boundary is not Loewner for each metric which is quasisymmetric to a Gromov type metric ln BourdoniPajot examples7 Patterson Sullivan measure works because there are relatively few quasisymmetric maps Problem 45 Kim Ruane Can you do analysis on CAT0 boundaries With no natural metric7 is there any structure beyond topology Remark 16 Bruce Kleiner Pushing in77 the visual sphere gives pseudo metrics on dooX where X is the CAT0 space acted on by G Consider the radial projection dooX a 5130 to spheres of radius R then dR Pr 1dXSR0 are the pseudo metrics But then for a function b going quickly enough to zero7 Z ltRgtdR BEN is a metric on dooX 14 MISHA KAPOVICH Remark 17 Damian Osajda One can de ne a family dA of metrics on ooX as follows Pick A gt 0 and choose a base point 0 E X Let 046 be geodesic rays emanating from 0 and asymptotic to points 7 77 E dooX Let a be such that daa7 a A lf such a does not exists ie Oz B then set a oo Finally7 set dAlt 7 77 Problem 46 Bruce Kleiner G lsomX acts on 300X ls this action nice77 with respect to the metrics in the previous remark a Problem 47 Marc Bourdon If D is the boundary of a hyperbolic group and D is connected7 has no local cut points7 and is not Loewner7 is there a quasisymmetrically invariant nontrivial closed equivalence relation on D so that D N is Hausdorff and is a boundary of G relative to a collection of parabolic subgroups Remark 18 In BourdoniPajot examples 117 the answer to the above problem is positive Problem 48 Jeremy Tyson Study relationships between different notions of con formal structure on 300639 for hyperbolic G Here is an incomplete list of such notions 1 1 quasiconformal in the metric sense7 ie Hf 1 2 preserving modulus of curves joining two compacta 3 77 quasisymmetric with 77 as close to linear as we like 77 are functions of the point z E 300G where we test f for conformality 4 if Poincare inequality holds for 300G then7 using Cheeger cotangent bundle T600G7 can give a notion of measurable bounded conformal structure 77 such that Conf600G7 77 is a convergence group Remark 19 Good notions of quasiconformality should have the convergence property7 and metric notion does not7 so its usefulness would be if 1 gt 27 since 1 is checkable and 2 is not Recall that if f R a R is a homeomorphism for 77 2 27 then k 7 quasiconformal gt quasisymmetric gt balls gt gt quasiballs Problem 49 Juha Heinonen ls the same true for Hilbert spaces All known proofs of above type use geometry7 not analysis It follows from work of Mario Bonk and Oded Schramm that there are quasi isometric embeddings of llllllll quaternionic hyperbolic space into llllllllm which are very far from isometric embeddings One can construct such examples with m c77 for a constant c which is no less than 16 PROBLEM LIST 15 Problem 50 David Fisher Can one do this with smaller m7 Say m n 1 Same problem valid for complex hyperbolic space Subproblem Misha Kapovich Consider X dool ll l sitting inside of Y QWHlllln ls X locally quasi symmetrically rigid in Y More precisely is it true that each quasisymmetric embedding f X a Y which is suf ciently close to the identity is induced by an isometry of HEW Remark 20 This subproblem might be easier to settle than Problem 50 since one can try to use in nitesimal tools like quasiconformal vector elds David Fisher and Kevin Whyte have constructed some exotic77 quasi isometric embeddings for higher rank symmetric spaces that are algebraic7 in the sense that 7T3A1N1 A2N2 Problem 51 David Fisher Are all quasi isometric embeddings between higher rank symmetric spaces either isometries or algebraic in this way Problem 52 Misha Kapovich Let G be a hyperbolic group ls it true that G admit a uniformly quasiconformal discrete action on S for some n The answer is probably negative It is reasonable to expect that every group satisfying Property T which admits such an action must be nite However the usual proofs that in nite discrete subgroups of lsomll 1 never satisfy Property T do not work in the quasiconformal category 9 PROBLEMS RELATED TO CANNON7S CONJECTURE Problem 53 Cannon7s Conjecture Version I If G is a Gromov hyperbolic group with 300G homeomorphic to 82 then G acts geometrically on H3 Problem 54 Cannon7s Conjecture Version ll Under the same assumptions on G 600Gvisual metric qu Szstandard Remark 21 Perelman7s proof of Thurston7s geometrization conjecture implies that the Cannon7s conjecture is equivalent to the nding that such G is commensurable to a 3 manifold group There exists an exact sequence l F GO W1M3 gt1 with F nite and G G0 lt 00 Remark 22 If G is hyperbolic and torsion free then 300G E 52 iff G is a PD3 group a 3 dimensional Poincare duality group see Problem 55 Conjecture of CTC Wall Every PD3 group is a 3 manifold group Problem 56 Cannon7s Conjecture Relative Version I If G is hyperbolic and 300G is homeomorphic to the Sierpinski carpet then G acts geometrically on a convex subset of H3 Remark 23 This follows from the Cannon7s Conjecture by doubling 16 MISHA KAPOVICH Problem 57 Cannon7s Conjecture Relative Version H For G hyperbolic relative to H1 Hk for a collection of virtually Z2 subgroups Hi if the Bowditch boundary QBOWG is 82 then G is commensurable to the fundamental group of a hyperbolic 3 manifold of nite volume Remark 24 The same problem could be posed allowing the boundary to be S2 or Sierpinski carpet Problem 58 Cannon7s Conjecture Analytic Version If G is a hyperbolic group with 300G homeomorphic to the Sierpinski carpet then the visual metric on 300G is quasisymmetric to some round Sierpinski metric 600Gvisual metric qu Sierpinskiround Recent work of Mario Bonk gives simpli cations and partial answers here Problem 59 Prove Cannon7s conjecture under additional assumptions such as o G 7T1M3 in Haken case without appealing to Thurston7s proof of the hyperbolization theorem 0 G a PD3 group that splits over a surface group 0 acts on a CAT0 cube complex 0 Remark 25 Cannon7s Conjecture is known for Coxeter groups G a work by Mario Bonk and Bruce Kleiner This follows of course from Andreev7s theorem but the point here is to give a proof which only uses the geometry of the ideal boundary of Problem 60 Misha Kapovich Give positive solution to Problem 57 assuming the absolute case by doing hyperbolic Dehn surgery77 see GrovesiManning Osin Namely add relators 1 i 1 k where R 6 H For suf ciently long elements R the quotient G P ltlt R1 Rk gtgt are known to be hyperbolic a Prove that if Rs are suf ciently long then G is an absolute PD3 group by say computing HG ZG b Assuming that each G is a 3 manifold group show that P is a 3 manifold group as well To motivate a possible approach to Cannon7s conjecture recall the following Theorem 2 Bonk B Kleiner Suppose that G is a hyperbolic group G n Z is a uniformly quasi Moebius action on a metric space which is topologically conjugate to the action of G on its ideal boundary Assume that Z is Ahlfors n regular and has topological dimension n Then Z is quasi symmetric to the round n sphere In particular G acts geometrically on Hn Therefore given a hyperbolic group G with Z 300G homeomorphic to S2 one would like to replace the visual metric d on 300G with a quasisymmetrically equivalent one which has Hausdorff dimension 2 Since Hausdorff dimension Hdim of a metric compact homeomorphic to S2 is 2 2 one could try to minimize Hausdoi dimension in the quasi conformal gauge of Z d ie the collection QZd of metric spaces Z d which are quasisymmetric to Z d This motivates the following PROBLEM LIST 17 De nition 3 For a metric space Z de ne its Pansu conformal dimension PGDZ infHdz39mY Y E QZ Likewise Ahlfors regular Pansu conformal dimension of X is AGDZ infHdz39mY Y E 9Z Y is Ahlfors regular The importance of the latter comes from Theorem 3 Bonk B Kleiner Suppose that G is a hyperbolic group Z 300G is homeomorphic to 82 If the AGDZ is attained then G acts geometrically on H3 Remark 26 BourdoniPajot examples 11 show that the AGD for the boundaries of hyperbolic groups is not always attained Generally AGDZ attained iff there is a Loewner metric in 9Z which is then minimizing Problem 61 Conjecture of Bruce Kleiner For a hyperbolic group G AGD600G GianJinz39mwcoXvisual where the in mum is taken over all geometric actions of G on metric spaces X A bolder conjecture would be that when the in mum is attained it is attained by a visual metric Problem 62 What is AGD of the standard Sierpinski carpet In particular does the above conjecture hold Problem 63 Juha Heinonen Under what assumptions on hyperbolic groups G with Q Loewner boundary 300G does it admit a 1 Poincare inequality for the boundary Cannon7s conjecture has a generalization to nonuniform convergence group actions on compacts Here is one of such Suppose L is the support of a measured lamination on a surface S and S L consists of topological disks Lift this lamination to a lamination A in the unit disk D C 2 and de ne the following equivalence relation N 1 The closure of each component of D A in the closed disk D is an equivalence class 2 If y C A is a geodesic which is not on the boundary of a component of D A then the closure of y in D is an equivalence class The rest of the points of 52 are equivalent only to themselves Note that N is G invariant and that the equivalence classes are cells Therefore the quotient 82 N is homeomorphic to 2 and the group G acts on 82 by homeomorphisms One can check that this is a convergence group action More generally one can form an equivalence relation using a pair of transversal laminations and make the corresponding Giinvariant quotient Problem 64 CannoniThurston ls this action conjugate to a conformal action The situation here is in many ways more complicated than in Cannon7s conjec ture since there is no a priori a useful metric structure on Z 82 It is not even 18 MISHA KAPOVICH clear that there exists a Gromov hyperbolic space X with the ideal boundary Z so that the action G n Z extends to a uniformly quasi isometric quasi action G n X One can reformulate this problem using theory of Kleinian groups as follows According to the Ending Lamination Conjecture there exists a discrete embedding LG C Isomllll3 so that the ending lamination of G is L Problem 65 The limit set of the Kleinian group LG is locally connected In the presence of two geodesic laminations the limit set of LG is the entire 2 sphere so local connectedness is meaningless Then the correct reformulation is as follows Problem 66 ls there an equivariant continuous map called CannoniThurston map from the unit circle Sl the ideal boundary of G as an abstract group to 52 Then Problem 64 is equivalent to 66 Remark 27 Positive solution of Problem 66 is known in certain cases For instance Jim Cannon and Bill Thurston showed this for laminations which are stable for a pseudo Anosov homeomorphism Yair Minsky 45 proved this under the assumption that the injectivity radius of HSLG is bounded away from zero Curt McMullen proved this in the case when G is the fundamental group of once punctured torus of quadruply punctured sphere 44 A complete solution of this problem is claimed in the recent preprint of Mahan Mj Mahan Mitra 47 Problem 67 Mahan Mitra Let H C G be a hyperbolic subgroup of a hyperbolic group we do not assume that H is quasiconvex ls it true that there exists an equivariant continuous map acoH a 300G See 46 for partial results in this direction 10 POISSON BOUNDARY Problem 68 Vadim Kaimanovich What is the Poisson boundary of the free group with an arbitrary measure Let Y d be a metric space and let CY denote the space of continuous functions on Y equipped with the topology of uniform convergence on bounded subsets Fixing a basepoint y E Y the space Y is continuously injected into CY by ltIgt z gt gt dz 7 dzy If Y is proper then Y is compact The points on the boundary Y ltIgtY are called horofunctz39ons or Busemann functions Problem 69 Conjecture of Anders Karlsson There almost surely exists a horo function h such that naoo 1 1 77h n A 1m n where A lim dz0zn PROBLEM LIST 19 A theorem of Karlsson states that V6 gt 0 there exists a horofunction h6 such that A763 7h5xn Ae for all n 2 NE Remark 28 This works for any nitely generated group Problem 70 Anders Karlsson For any proper metric space it is possible to as sociate a kind of incidence geometry at in nity via horofunctions7 halfspaces and their limits called stars For the CAT0 case7 this structure is intimately connected with the Tits geometry7 and for Teichmiiller space it should relate well with the curve complex In which situations do homomorphisms induce incidence preserving77 maps between these geometries at in nity Same problem for quasi isometries Problem 71 Anders Karlsson Consider the compacti cation of a nitely generated group constructed in the usual Stone Cech way using the rst l2 or some other func tion space cohomology ls the associated incidence geometry at in nity always trivial ie7 hyperbolic This is related to problems of Gromov in Asymptotic invariants in the chapter on l cohomology Kaimanovich and Masur proved the following for a measure M on the mapping class group P M can be any nite rst moment7 nite entropy probability measure such that the group generated by its support is non elementary7 there exists a measure V on PM so that PoissPi PA1f V The measure V is called a ii statioriary measure an PMf this measure is unique Here Poissl7 M is the Poisson boundary Problem 72 Moon Duchin Characterize the hitting measure V on PM obtained from the random walk by mapping classes on Teichmiiller space ls it absolutely continuous with respect to visual measure that is7 Lebesgue measure on the visual sphere of directions Problem 73 Moon Duchin What is the Poisson boundary of Outer space 11 ASYMPTOTIC CONES A geodesic metric space X eg Cayley graph of a nitely generated group is Gromov hyperbolic if and only if all asymptotic cones of X are trees There are examples of nitely generated 1 groups G so that some asymptotic cones of G are trees but G is not Gromov hyperbolic7 see 58 Call such groups laeuriary hyperbolic following Olshansky7 Osin and Sapir see 49 All such groups are limits of hyperbolic groups in the sense that G admits an in nite presentation G lt 17 7an17R27R37 gt so that each Gk lt 17zan17Rk gt is hyperbolic Problem 74 Misha Kapovich ls there are meaningful structure theory for lacunary hyperbolic groups Can one de ne a useful boundary for such groups ls it true that either OutG is nite or G splits over a virtually cyclic subgroup 1Such groups are never nitelypresented 20 MISHA KAPOVICH Remark 29 A counter example to the last problem is known to M Sapir It is known that for each relatively hyperbolic group G7 all asymptotic cones of G have cut points Problem 75 Cornelia Drutu To what extent is the reverse implication true Remark 30 Some counterexamples are known for instance7 the mapping class group and fundamental groups of graph manifolds are weakly relatively hyperbolic but not strongly Problem 76 Mario Bonk The study of asymptotic cones has been non analytic they have been studied up to homeomorphism What analytic tools could be de veloped 12 KLEINIAN GROUPS Problem 77 Misha Kapovich For the fundamental group G of a closed hyperbolic n manifold consider a short exact sequence 1 a Z17 a P a G a l ls the group P residually nite In other words7 is there a nite index subgroup G in G so that the restriction map H2GZp a H2G Z is zero Remarkably7 positive answer is presently known only for n 2 Same problem makes sense also for the fundamental groups of complex hyperbolic and quaternionic hyperbolic manifolds Problem 78 Misha Kapovich Let G be as above ls there a nite index subgroup G C G so that the restriction map H3GZ2 a H3G Z2 is zero This problem is interesting because H3GZ2 classi es PL structures on the hy perbolic manifold llllnG Problem 79 Misha Kapovich Let G be a Gromov hyperbolic Coxeter group Does G admit a discrete embedding in Isomllll for large 71 Note that the Coxeter generators are not assumed to act as re ections on H Otherwise7 there are counter examples7 see 31 Problem 80 Misha Kapovich Let G C PU27 1 be a convex cocompact subgroup of isometries of complex hyperbolic 2 space Can the limit set of G be homeomorphic to the Sierpinski carpet Problem 81 Misha Kapovich Let G C Isomllll be a discrete torsion free nitely generated subgroup without abelian subgroups of rank 2 2 ls it true that a chG S Hdz39mAcG 1 PROBLEM LIST 21 Here Ac is the conical limit set The answer is known 36 to be positive if one considers homological rather than cohomological dimension b In the case of equality7 is it true that the limit set of G is the round sphere and G This is known to be true in the case when G is geometrically nite 36 c If Hdz39mAcG lt 27 is it true that G is geometrically nite d If Hdz39mAcG lt 17 does it follow that G is a classical Schottky type group le the one whose fundamental domain is bounded by round spheres See 34 for partial results Problem 82 Lewis Bowen Let G C Isomllll4 be a Schottky group or7 more generally7 a free convex cocompact group Can Hausdorff dimension of the limit set of G be arbitrarily close to 3 Problem 83 Misha Kapovich Let G be a nitely generated discrete group of isometries of a Gromov hyperbolic space X so that the limit set of G is connected ls it true that the limit set of G is locally connected Consider a representation p G a lsomlHl This action of G on the hyperbolic space determines a class function I G a R so that 6A9 is the displacement for the isometry pg of llll 7 ie7 My mg dP9957 96 Problem 84 Suppose that phpg are discrete and faithful representations as above so that there exists G gt 0 for which we have 0 1M30 VgEG 4229 Does it follow that there exists a quasiconformal map f Ap1G a Ap2G which is equivariant with respect to the isomorphism pg 0 pfl Can one choose f which is K quasiconformal for K KG If n 3 and G is nitely generated7 then the answer to the rst part of the problem is positive and follows from the solution of the ending lamination conjecture A constructive proof of Rips compactness theorem Let G be a nitely presented group which does not split as a graph of groups with virtually abelian edge groups For every 71 de ne the space DAG of conjugacy classes of discrete and faithful representations of G into lsomlHl We assume that G is not virtually abelian itself Then Rips7 theory of group actions on trees implies that DAG is compact Problem 85 Find a constructive proof of the above theorem More precisely7 consider a nite presentation lt91gklR1 Rm of G Given p E DAG de ne EndJD 11 jgka dp9ix Find an explicit constant G7 which depends on 717 gm and the lengths of the words Ri so that the function Bn DAG a R is bounded from above by G 22 MISHA KAPOVICH Remark 1 Y Lai 41 found such an explicit constant C can be for Coxeter groups moreover in this case G depends only on n and the number of Coxeter generators One possible application of the solution of Problem 85 is in producing nontrivial algebraic restrictions on Kleinian groups An abstract Kleinz39an group is a group which admits a discrete embedding in lsomllll for some n All currently known algebraic restrictions on nitely generated Kleinian groups can be traced to the following 1 Every Kleinian group has the Haagerup property They admit isometric prop erly discontinuous actions on some Hilbert space See for instance 19 2 If 7139 is the fundamental group of a compact Kahler manifold then every ho momorphism p 7r a lsomllll wither factors through a group commensurable to a surface group or p7r preserves a point or a pair of points in l l U 600M See 17 Problem 86 Find new restrictions on Kleinian groups Recall that a group G is called coherent if every nitely generated subgroup of G is nitely presented Problem 87 Kapovich L Potyagailo EB Vinberg Prove that every arith metic lattice in lsoml l n 2 4 is non coherent See 38 for some partial results in this direction It is well known that every lattice in lsomllllllll n 2 2 has Property T Problem 88 Suppose that G C lsomllllllll is a discrete subgroup satisfying Prop erty T Does it follow that G preserves a totally geodesic subspace H in l ll l and acts on H as a lattice The main motivation for this problem comes from the fact that the obvious con strictions of discrete groups of isometries are by various graphs of groups and hence these groups do not have Property T One can try to use triangles of groups Problem 89 Suppose that A is a developable triangle of groups where all the cell groups have Property T and so that all the links in the universal cover of T have A1 gt 12 Does it follow that 7T1A has Property T Problem 90 Generalize Bestvina Feighn combination theorem from graphs of gro ups to complexes of groups Background Let g be a graph of groups so that vertex and edge groups are hyperbolic and the edge subgroup are quasiconvex in the vertex groups Bestvina and Feighn 5 found some suf cient conditions for Mg to be hyperbolic Hammenstadt has some partial results towards solving this problem Discrete subgroups in other Lie groups A reflection in a complex hyperbolic space CH is an isometry of nite order which xes a complex codimension 1 hyperplane A reflection group in Illll is a subgroup of PUn 1 generated by re ections These concepts generalize the notion of re ections and re ection groups acting on H Vinberg 60 proved that there for PROBLEM LIST 23 n 2 30 there are no uniform lattices in On1 which are re ection groups This result was extended by Prokhorov 52 who proved nonexistence of re ection lattices in On1 for n 2 996 Problem 91 Generalize Vinberg7s niteness theorem for re ection groups to com plex hyperbolic re ection groups7 ie7 prove that there exists a number N such that for n 2 N7 there are no lattices in PUn1 which are generated by re ections Problem 92 Misha Kapovich There is a theory of quasi convex groups acting on Gromov hyperbolic spaces7 generalizing the theory of convex compact groups of isornetries of the real hyperbolic space Develop a theory of geometric niteness in CAT0 spaces Remark 31 It is a priori unclear what to take as the de nition of geometric niteness in the context of CAT0 spaces even in the case of symmetric spaces Taking quotients of the convex hull is a bad idea7 as shown by a theorem of Bruce Kleiner and Bernhard Leeb There are only few convex subsets in symmetric spaces of rank gt 2 A better de nition replacing convex cocompactness could be A nitely generated group G C lsomX is undistorted if the induced map from the Cayley graph of G to X is a quasi isometric embedding In the case of Gromov hyperbolic spaces7 undistorted is equivalent to quasi convent There are examples of undistorted free Zariski dense subgroups of SLnR7 gen eralizing the Schottky construction ls there an interpretation of the notion of undistorted groups in terms of the group actions on limit sets F Labourie 40 introduced another notion of convex cocompactness that he calls an Anosou structure7 for group representations p P a G7 where G is a semisimple Lie group In the case when P is a surface group and G SLn 17R7 this notion can be reformulated in terms of action of pP on its limit set in RP ie existence of a pP invariant hyperconvex curve in RP Problem 93 Anna Wienhard Extend this relation of Anosov structure and dy namics on the limit set to representations of other hyperbolic groups Problem 94 Anna Wienhard Generalize holomorphic chain patterns in QWCH in order to prove rigidity results for embeddings of lattices in PUn1 into other higher rank Lie groups Background ldeal boundaries of totally geodesic subspaces CH1 C CH de ne holomorphic chains in QWCH These circles are characterized by the property that three points belong to such a chain if and only if they span an ideal triangle in Illll of maximal symplectic area The incidence relation between holomorphic chains in QWCH determines a building like77 structure where chains serve as apartments Every two points belong to a chain Given a measurable map QWCH a 300Cllllmm 2 n7 24 MISHA KAPOVICH which induces a measurable morphism of these building like77 structures7 is induced by a holomorphic embedding CH a Cllllm This7 in turn7 can be used to reprove Cor lette7s rigidity theorem 21 for representations of lattices in PUn1 into PUm1 The motivation for the Problem 94 is to extend Corlette7s rigidity result to represen tations of PUn1 to other Lie groups Problem 95 Anna Wienhard Obtain new rigidity results for embeddings of real hyperbolic lattices into higher rank semisimple Lie groups in terms of the boundary maps 13 MISCELLANEOUS PROBLEMS IN GEOMETRIC GROUP THEORY Problem 96 Kevin Whyte Homotopy Nielsen realization If X is a compact poly hedron and G is a discrete group of simple homotopy equivalences X a X7 is there a compact space X 7 homotopy equivalent to X7 such that G can be realized as a group of homeomorphisms of X Remark 32 It is a long standing open problem to determine if the exact sequence 1 a H0me00S a H0me0S a M0dS a 1 is split Here S is a compact surface of genus 2 2 and H0me00S denotes the connected component of the identity in the group of homeomorphisms Moreover7 there are examples due to George Cooke of spaces X and nite groups G of simple homotopy equivalences of X for which the answer to Problem 96 is No77 However all known examples occur in dimensions 2 5 and require the group to have torsion Problem 97 llia Kapovich Consider nite cell complexes X ls there an algorithm to determine if X is contractible Remark 33 The triviality problem is known to be unsolvable for nitely presented groups However7 their presentation complexes are never contractible7 since the pre sentation is unbalanced On the other hand7 for complexes of dimension n 2 4 there is no algorithm to determine contractibility S Weinberger 62 lndeed7 take a trian gulated closed n manifold M for which it is impossible to decide if M is homeomorphic to S Let X be the complement to an open n simplex in M Then contractibility of X is undecidable7 since it is equivalent to M being a homotopy sphere Remark 34 Daniel Groves It is an open problem if the triviality of a group is algorithmically solvable for groups with balanced presentation A presentation is called balanced if the number of generators equals the number of relators Problem 98 Kevin Whyte For a word hyperbolic G not splitting over any vir tually cyclic group7 can an in nite index subgroup and a nite index subgroup be isomorphic Remark 35 This asks for something slightly stronger than the cohop an property Problem 99 Misha Kapovich Consider Teichmiiller space TS with Teichmiiller metric Does it have quadratic isoperimetric inequality PROBLEM LIST 25 Background lf dimCTS 2 27 TS is known to be non hyperbolic How ever the Mapping Class Group is bi automatic7 therefore the thick part77 of TS is semihyperbolic One can ask a similar question for the outer space Curt McMullen de ned in exibility for Kleinian groups7 43 Problem 100 Danny Calegari ls there a similar statement to this in exibility result this with no group speci edithat is7 for subsets A C 52 of the boundary sphere of H3 Here is a possible setup for such problem De ne a random Beltrami differential as follows Let 739 be the tessellation of H2 by regular right angled hyperbolic pentagons All such pentagons are isometric to a model pentagon P Let M be a compact perhaps nite or even a singleton set of Beltrami differentials on P having norm 12 or any xed number lt 1 For concreteness7 suppose that M no is a singleton For each pentagon P E 739 choose a random isornetry g P a P There are 10 such isornetries Then push forward 0 from P to P via 9 This de nes a random Beltrami di ereritial M on H2 Given a closed connected set A C 2 observe that each complementary component 9 C 2 A is simply connected Choose a Riemann mapping R H2 a Q push forward the random Beltrami differential M from H2 to 9 Repeat this for each component of 2 A and extend the resulting differential to A by zero This de nes a Beltrami di ereritial MA on 82 Let q qA 52 a 52 be the quasiconformal map which is a solution of the Beltrami equation Q2 MAQZ39 Such a quasiconformal map has a natural Thurston Reimarm biLipschitZ extension QA lll13 a llllil7 53 Problem 101 Given p E llllzl7 estimate the biLipschitZ constant of QA near p in terms of the distance d from p to the exterior of the convex hull of A More concretely if A is a quasicircle7 is the decay exponential in d That is7 are there positive constants 0102 such that LltQA7P S 1 015p702dp7600K where L is the bilipschitz constant of QA restricted to the ball of some xed radius say radius 1 about p7 K is the convex hull of A7 and p is a point in the interior of K Problem 102 Mladen Bestvina Are braid groups CAT07 Remark 36 It is conjectured that all Artin groups are CAT0 Problem 103 Extend Rips7 theory to higher dimensional buildings7 eg products of R trees Rank rigidity Let X be a CAT0 metric space The space X is said to be or rank 2 n if every geodesic segment in X is contained in a subset E which is isometric to a at n dimensional parallelepiped If Y is a locally CAT0 metric space7 then Y is said to have rank 2 n if its universal cover is of rank 2 n The rank rigidity theorem proven by Ballmann 1 and by Burns and Spatzier 157 16 states that 26 MISHA KAPOVICH If M is a compact nonpositively curved Riemannian manifold of rank 2 27 then either M admits a nite cover the universal cover of M splits nontrivially as a Riemannian direct product or M is a locally symmetric space Problem 104 Werner Ballmann7 Misha Brin Suppose that Y is a compact nite dimensional locally GAT0 metric space of rank n 2 2 Then either the universal cover of Y splits nontrivially as a Riemannian direct product or it is isometric to a Euclidean building This problem is most natural in the context of piecewise Euclidean metric cell complexes The conjecture was proven in the case of 2 dimensional and 3 dimensional complexes by Ballmann and Erin 27 3 Cogrowth Let H C G be a subgroup of a nitely generated group G The cogrowth of H in G is the growth of the Shreier graph l GH7 where PC is a Cayley graph of G Problem 105 Compute cogrowth for interesting77 subgroups For instance 1 Show that the cogrowth of SLn7 Z in SLn 17 is exponential 2 Compute cogrowth of special subgroups in Coxeter groups See 61 for partial results 3 Suppose that PaH is Gromov hyperbolic ls it true that the cogrowth is either constant7 linear or exponential Coarse Whitehead Conjecture Problem 106 Whitehead Conjecture Let X be an aspherical ie with con tractible universal cover 2 dimensional complex ls it true that every subcomplex of X is also aspherical See A metric space Z is said to be coarsely trivial Wm if the following holds There exists a function R so that for each R 2 0 the map R p5RZ a R p5 RZ induces zero map of the m th homotopy groups For instance7 suppose that Z is the 0 skeleton of an m connected simplicial complex X7 which admits a cocompact free group action Metrize Z by declaring each edge of X to have unit length Then Z has coarsely trivial 7g for all j S m Given a 2 dimensional contractible complex X as above and a connected subgraph Y C X 7 metrize Y0 using the above path metric on Y Problem 107 Coarse Whitehead Conjecture 7 Misha Kapovich Under the above assumptions7 is it true that Y has coarsely trivial Wm for m 2 2 More restrictively one can consider the case when X is the Cayley complex of a nitely presented group G and H is nitely generated subgroup of G7 identi ed with Y0 Then the metric on Y0 is quasi isometric to the word metric on Note that if H is nitely presented then since its 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