Mathematics Problem Archive

Suppose that Z is a compact metrizable topological space, G↷ Z is a convergence action which is topologically transitive, i.e. each G– orbit is dense ...

Find topological restrictions on the ideal boundaries of CAT (−1) cubical complexes....

Is it true that isomorphic Coxeter groups have homeomorphic boundaries?...

Does there exist a Coxeter group Gn with n-dimensional boundary ∂Gn, so that the rational homological dimension of ∂Gn equals 1?...

Under which conditions on the Coxeter diagram of G, the boundary of a Coxeter group is n-connected and locally n-connected?...

Can exotic homology manifolds as in [14] appear as ideal boundaries of Coxeter groups?...

Find further universality phenomena: classes of groups or spaces of different nature whose ideal boundaries are nevertheless all homeomorphic, beyond ...

Let $N$ be a closed $n$-manifold and $\Delta$ a flag, no-square triangulation, and let $C(N,\Delta)$ be the associated Davis--Vinberg complex. Is $\pa...

Suppose that ( N1, ∆1) and (N2, ∆2) are closed 3-manifolds equipped with flag-triangulations, so that ∂∞C(N1, ∆1) = ∂∞C(N2, ∆2). Does it follow that ev...

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 compactum, so...

Examples of Kleiner and Croke [22], [23] of non-unique boundaries are badly non-locally-connected. Is that essential in having the “flexibility” to hav...

Suppose that G is a CAT (0) group which does not split over a small subgroup. Does it follow that ∂∞G is unique?...

Is the boundary well-defined for groups acting geometrically on CAT (0)-cube complexes? More precisely, suppose that X1, X2 are cube complexes which ad...

What topological invariants distinguish boundaries? In particular, what topological properties of boundaries are quasi-isometry invariants? Does somet...

If 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...

Is 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...

Define 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...

Any 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...

Let 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?...

Can there be two different boundaries in the sense of Z-structures for a group G that are not cell-like equivalent?...

Is the property of splitting over a 2-ended subgroup an invariant of Bestvina boundaries? Some necessary conditions are known for compact, metrizable ...

Extend 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...

Does every CAT(0) group have finite asymptotic dimension?...

Do Papasoglu's three results for finitely presented one-ended groups extend to all finitely generated groups: quasi-isometry invariance of the JSJ dec...

Are splittings overZ2 (orZn) invariant under quasiisometry? The analogous problem also makes sense for the JSJ decompositions....

Suppose G is finitely generated and there is a sequence of quasicircles that separate its Cayley graph. Is G virtually a surface group?...

If 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...

Do all homogeneous continua (with dimension greater than 2) have this property?...

Are diffeomorphisms Rn →Rn dense in the space of all quasiconformal maps?...

Let f: Bn → Bn be a quasiconformal homeomorphism. Can f be approximated by globally quasiconformal diffeomorphisms fj: Bn → Bn? Can this be done so tha...

Find good classes of spaces such that the infinitesimal metric condition (for quasiconformality) implies the local condition. (This is generally true i...

Outside of the boundaries of Fuchsian buildings, what boundaries have the Loewner property?...

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 op...

Develop a theory for analysis on the ideal boundaries of relatively hyperbolic groups, as it is done for hyperbolic groups....

In what generality does quasiconformal imply quasisymmetric?...

Take your favorite metric fractal. Is it quasisymmetrically cohopfian? What about the boundaries of hyperbolic groups?...

If ∂∞G is Loewner, then it is quasisymmetrically cohopfian. (Boundaries of Fuchsian buildings provide a good test case for this conjecture.)...

If G is a hyperbolic group and ∂∞G is connected with no local cut points, is there a natural measure class which is quasisymmetrically invariant?...

Can you do analysis on CAT(0) boundaries? With no natural metric, is there any structure beyond topology?...

G = Isom( X) acts on ∂∞(X). Is this action “nice” with respect to the metrics in the previous remark?...

If 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...

Study relationships between different notions of conformal structure on ∂∞(G) for hyperbolic G. Here is an (incomplete) list of such notions: (1) 1-qua...

For homeomorphisms of Hilbert spaces, do the Euclidean implications 'quasiconformal implies quasisymmetric implies mapping balls to quasiballs' contin...

Can one do this with smaller m? Say m = n + 1? (Same problem valid for complex hyperbolic space.) Subproblem (Misha Kapovich): Consider X = ∂∞HHn sitt...

Are all quasi-isometric embeddings between higher-rank symmetric spaces either isometries or algebraic in this way?...

Let G be a hyperbolic group. Is it true that G admit a uniformly quasiconformal discrete action on Sn (for some n)?...

For 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...

What is ACD of the standard Sierpinski carpet? In particular, does the above conjecture hold?...

Under what assumptions on hyperbolic groups G with Q-Loewner boundary ∂∞G does it admit a 1-Poincar´ e inequality for the boundary?...

[Cannon–Thurston] Is this action conjugate to a conformal action?...