Mathematics Problem Archive

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 ...

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...

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...

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?...

The 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...

Is 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...

For any proper metric space it is possible to associate a kind of incidence geometry at infinity via horofunctions, halfspaces and their limits called ...

Consider the compactification of a finitely generated group constructed in the usual Stone- ˇCech way using the first l2 (or some other function space) c...

The study of asymptotic cones has been non-analytic (they have been studied up to homeomorphism). What analytic tools could be developed?...

Let $G$ be the fundamental group of a closed hyperbolic $n$-manifold. Is there a finite-index subgroup G′ ⊂ G so that the restriction map H 3(G,Z2) → H...

Let G ⊂ P U(2, 1) be a convex-cocompact subgroup of isometries of complex-hyperbolic 2-space. Can the limit set of G be homeomorphic to the Sierpinski...

Let G ⊂ Isom(H4) be a Schottky group (or, more generally, a free convex-cocompact group). Can Hausdorff dimension of the limit set of G be arbitrarily ...

Let G be a finitely-generated discrete group of isometries of a Gromov-hyperbolic space X so that the limit set of G is connected. Is it true that the ...

Let $G$ be a group and let $\rho_1,\rho_2:G\to\operatorname{Isom}(\mathbb{H}^n)$ be discrete faithful representations; write $\ell_{\rho}(g)=\inf_x d(...

Find a “constructive” proof of the above theorem. More precisely, consider a finite presentation ⟨g1,.., gk|R1,.., Rm⟩ of G. Given [ ρ] ∈ D n(G) define ...

Find new restrictions on Kleinian groups. Recall that a group G is called coherent if every finitely-generated subgroup of G is finitely-presented....

Prove that every arithmetic lattice in Isom( Hn) ( n ≥ 4) is non-coherent. See [38] for some partial results in this direction. It is well-known that ...

Suppose that G ⊂ Isom(HHn) is a discrete subgroup satisfying Property T. Does it follow that G preserves a totally-geodesic subspace H in HHn and acts...

Suppose that ∆ is a developable triangle of groups, where all the cellgroups have Property T and so that all the links in the universal cover of T hav...

Generalize Vinberg’s finiteness theorem for reflection groups to complex-hyperbolic reflection groups, i.e., prove that there exists a number N such that...

Generalize holomorphic chain patterns in ∂∞CHn in order to prove rigidity results for embeddings of lattices in P U(n, 1) into other higher rank Lie g...