Mathematics Problem Archive
Boundaries of Groups and Kleinian Groups — Problem 20
v1.3 research notesWhat topological invariants distinguish boundaries? In particular, what topological properties of boundaries are quasi-isometry invariants? Does somet...
Boundaries of Groups and Kleinian Groups — Problem 21
v1.3 research notesIf G acts geometrically on two CAT(0) spaces, are the resulting boundaries cell-like equivalent? (That is, does there exist a space Z with cell-like m...
Boundaries of Groups and Kleinian Groups — Problem 22
v1.3 research notesIs there a convex core for the diagonal action of G on X1 × X2? (A special case is surface groups G with X1 and X2 corresponding to different hyperboli...
Boundaries of Groups and Kleinian Groups — Problem 23
v1.3 research notesDefine a topology on the set of quasi geodesics in a (proper geodesic, or coarsely homogeneous, or cocompact) CAT (0) space which (1) has a description...
Boundaries of Groups and Kleinian Groups — Problem 24
v1.3 research notesAny CAT (0) group has no infinite-torsion subgroups. A Euclidean retract is a compact space that embeds into some Rn as a retract. A compact metrizable...
Boundaries of Groups and Kleinian Groups — Problem 25
v1.3 research notesLet G be a hyperbolic group and ∂EZ G be its EZ boundary. Is it true that ∂EZ G is equivariantly homeomorphic to the Gromov boundary of G?...
Boundaries of Groups and Kleinian Groups — Problem 26
v1.3 research notesCan there be two different boundaries in the sense of Z-structures for a group G that are not cell-like equivalent?...
Boundaries of Groups and Kleinian Groups — Problem 27
v1.3 research notesIs the property of splitting over a 2-ended subgroup an invariant of Bestvina boundaries? Some necessary conditions are known for compact, metrizable ...
Boundaries of Groups and Kleinian Groups — Problem 28
v1.3 research notesExtend the known necessary conditions for a compact metrizable space $X$ to be the boundary of a proper cocompact CAT(0) space, or give a complete cla...
Boundaries of Groups and Kleinian Groups — Problem 29
v1.3 research notesDoes every CAT(0) group have finite asymptotic dimension?...
Boundaries of Groups and Kleinian Groups — Problem 30
v1.3 research notesDo Papasoglu's three results for finitely presented one-ended groups extend to all finitely generated groups: quasi-isometry invariance of the JSJ dec...
Boundaries of Groups and Kleinian Groups — Problem 31
v1.3 research notesAre splittings overZ2 (orZn) invariant under quasiisometry? The analogous problem also makes sense for the JSJ decompositions....
Boundaries of Groups and Kleinian Groups — Problem 32
v1.3 research notesSuppose G is finitely generated and there is a sequence of quasicircles that separate its Cayley graph. Is G virtually a surface group?...
Boundaries of Groups and Kleinian Groups — Problem 33
v1.3 research notesIf G is finitely generated with asymptotic dimension ≥ n, and X is a subset of the Cayley graph with asymptotic dimension ≤ n − 2 that coarsely separat...
Boundaries of Groups and Kleinian Groups — Problem 34
v1.3 research notesDo all homogeneous continua (with dimension greater than 2) have this property?...
Boundaries of Groups and Kleinian Groups — Problem 35
v1.3 research notesAre diffeomorphisms Rn →Rn dense in the space of all quasiconformal maps?...
Boundaries of Groups and Kleinian Groups — Problem 36
v1.3 research notesLet f: Bn → Bn be a quasiconformal homeomorphism. Can f be approximated by globally quasiconformal diffeomorphisms fj: Bn → Bn? Can this be done so tha...
Boundaries of Groups and Kleinian Groups — Problem 37
v1.3 research notesFind good classes of spaces such that the infinitesimal metric condition (for quasiconformality) implies the local condition. (This is generally true i...
Boundaries of Groups and Kleinian Groups — Problem 38
v1.3 research notesOutside of the boundaries of Fuchsian buildings, what boundaries have the Loewner property?...
Boundaries of Groups and Kleinian Groups — Problem 39
v1.3 research notesLet X be a non-smoothable closed simply connected 4-manifold. Does it admit an Ahlfors 4-regular linearly locally contractible metric? This is wide op...
Boundaries of Groups and Kleinian Groups — Problem 40
v1.3 research notesDevelop a theory for analysis on the ideal boundaries of relatively hyperbolic groups, as it is done for hyperbolic groups....
Boundaries of Groups and Kleinian Groups — Problem 41
v1.3 research notesIn what generality does quasiconformal imply quasisymmetric?...
Boundaries of Groups and Kleinian Groups — Problem 42
v1.3 research notesTake your favorite metric fractal. Is it quasisymmetrically cohopfian? What about the boundaries of hyperbolic groups?...
Boundaries of Groups and Kleinian Groups — Problem 43
v1.3 research notesIf ∂∞G is Loewner, then it is quasisymmetrically cohopfian. (Boundaries of Fuchsian buildings provide a good test case for this conjecture.)...
Boundaries of Groups and Kleinian Groups — Problem 44
v1.3 research notesIf G is a hyperbolic group and ∂∞G is connected with no local cut points, is there a natural measure class which is quasisymmetrically invariant?...
Boundaries of Groups and Kleinian Groups — Problem 45
v1.3 research notesCan you do analysis on CAT(0) boundaries? With no natural metric, is there any structure beyond topology?...
Boundaries of Groups and Kleinian Groups — Problem 46
v1.3 research notesG = Isom( X) acts on ∂∞(X). Is this action “nice” with respect to the metrics in the previous remark?...
Boundaries of Groups and Kleinian Groups — Problem 47
v1.3 research notesIf D is the boundary of a hyperbolic group and D is connected, has no local cut points, and is not Loewner, is there a quasisymmetrically invariant no...
Boundaries of Groups and Kleinian Groups — Problem 48
v1.3 research notesStudy relationships between different notions of conformal structure on ∂∞(G) for hyperbolic G. Here is an (incomplete) list of such notions: (1) 1-qua...
Boundaries of Groups and Kleinian Groups — Problem 49
v1.3 research notesFor homeomorphisms of Hilbert spaces, do the Euclidean implications 'quasiconformal implies quasisymmetric implies mapping balls to quasiballs' contin...
Boundaries of Groups and Kleinian Groups — Problem 50
v1.3 research notesCan one do this with smaller m? Say m = n + 1? (Same problem valid for complex hyperbolic space.) Subproblem (Misha Kapovich): Consider X = ∂∞HHn sitt...
Boundaries of Groups and Kleinian Groups — Problem 51
v1.3 research notesAre all quasi-isometric embeddings between higher-rank symmetric spaces either isometries or algebraic in this way?...
Boundaries of Groups and Kleinian Groups — Problem 52
v1.3 research notesLet G be a hyperbolic group. Is it true that G admit a uniformly quasiconformal discrete action on Sn (for some n)?...
Boundaries of Groups and Kleinian Groups — Problem 61
v1.3 research notesFor a hyperbolic group G, ACD (∂∞G) = inf G↷X {Hdim(∂∞X, visual)}, where the infimum is taken over all geometric actions of G on metric spaces X. A bol...
Boundaries of Groups and Kleinian Groups — Problem 62
v1.3 research notesWhat is ACD of the standard Sierpinski carpet? In particular, does the above conjecture hold?...
Boundaries of Groups and Kleinian Groups — Problem 63
v1.3 research notesUnder what assumptions on hyperbolic groups G with Q-Loewner boundary ∂∞G does it admit a 1-Poincar´ e inequality for the boundary?...
Boundaries of Groups and Kleinian Groups — Problem 64
v1.3 research notes[Cannon–Thurston] Is this action conjugate to a conformal action?...
Boundaries of Groups and Kleinian Groups — Problem 65
v1.3 research notesThe limit set of the Kleinian group ι(G) is locally connected. In the presence of two geodesic laminations, the limit set of ι(G) is the entire 2-sphe...
Boundaries of Groups and Kleinian Groups — Problem 66
v1.3 research notesIs there an equivariant continuous map (called Cannon–Thurston map) from the unit circle S1 (the ideal boundary of G as an abstract group) to S2? Then...
Boundaries of Groups and Kleinian Groups — Problem 67
v1.3 research notesLet H ⊂ G be a hyperbolic subgroup of a hyperbolic group (we do not assume that H is quasiconvex). Is it true that there exists an equivariant continu...
Boundaries of Groups and Kleinian Groups — Problem 68
v1.3 research notesWhat is the Poisson boundary of the free group with an arbitrary measure?...
Boundaries of Groups and Kleinian Groups — Problem 69
v1.3 research notesThere almost surely exists a horofunction h such that lim n→∞ − 1 n h(xn) = A, where A = lim n→∞ 1 n d(x0, xn). A theorem of Karlsson states that ∀ϵ >...
Boundaries of Groups and Kleinian Groups — Problem 70
v1.3 research notesFor any proper metric space it is possible to associate a kind of incidence geometry at infinity via horofunctions, halfspaces and their limits called ...
Boundaries of Groups and Kleinian Groups — Problem 71
v1.3 research notesConsider the compactification of a finitely generated group constructed in the usual Stone- ˇCech way using the first l2 (or some other function space) c...
Boundaries of Groups and Kleinian Groups — Problem 72
v1.3 research notesCharacterize the hitting measure ν on PMF obtained from the random walk by mapping classes on Teichm¨ uller space. Is it absolutely continuous with re...
Boundaries of Groups and Kleinian Groups — Problem 73
v1.3 research notesWhat is the Poisson boundary of Outer space?...
Boundaries of Groups and Kleinian Groups — Problem 74
v1.3 research notesIs there are meaningful structure theory for lacunary hyperbolic groups? Can one define a useful boundary for such groups? Is it true that either Out(G...
Boundaries of Groups and Kleinian Groups — Problem 75
v1.3 research notesEvery relatively hyperbolic group has cut points in all of its asymptotic cones. To what extent does the converse hold? Characterize the finitely gene...
Boundaries of Groups and Kleinian Groups — Problem 76
v1.3 research notesThe study of asymptotic cones has been non-analytic (they have been studied up to homeomorphism). What analytic tools could be developed?...
Boundaries of Groups and Kleinian Groups — Problem 77
v1.3 research notesFor the fundamental group G of a closed hyperbolic n-manifold consider a short exact sequence 1 →Zp → Γ → G → 1. Is the group Γ residually finite? In o...