Mathematics Problem Archive

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?...

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...

Let 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...

What is the Poisson boundary of the free group with an arbitrary measure?...

There 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 >...

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...

Characterize the hitting measure ν on PMF obtained from the random walk by mapping classes on Teichm¨ uller space. Is it absolutely continuous with re...

What is the Poisson boundary of Outer space?...

Is 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...

Every relatively hyperbolic group has cut points in all of its asymptotic cones. To what extent does the converse hold? Characterize the finitely gene...

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

For 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...

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 be a Gromov-hyperbolic Coxeter group. Does G admit a discrete embedding in Isom(Hn) for large n?...

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(Hn) be a discrete torsion-free finitelygenerated subgroup without abelian subgroups of rank ≥ 2. Is it true that (a) cdZ(G) ≤ Hdim(Λc(G)) ...

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 Bestvina-Feighn combination theorem from graphs of groups to complexes of groups....

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

There is a theory of quasi-convex groups acting on Gromov hyperbolic spaces, generalizing the theory of convex-compact groups of isometries of the rea...

Extend this relation of Anosov structure and dynamics on the limit set to representations of other hyperbolic groups....

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...

Obtain new rigidity results for embeddings of realhyperbolic lattices into higher-rank semisimple Lie groups in terms of the boundary maps....

If X is a compact polyhedron and G is a discrete group of simple homotopy equivalences X → X, is there a compact space X ′, homotopy equivalent to X, ...

Consider finite cell complexes X. Is there an algorithm to determine if X is contractible?...

For a word-hyperbolic G not splitting over any virtually cyclic group, can an infinite-index subgroup and a finite-index subgroup be isomorphic?...

Consider Teichm¨ uller spaceT (S) with Teichm¨ uller metric. Does it have quadratic isoperimetric inequality?...

Is there a similar statement to this inflexibility result this with no group specified—that is, for subsets Λ ⊂ S2 of the boundary sphere of H3?...

Given p ∈ H3, estimate the biLipschitz constant of QΛ near p in terms of the distance d from p to the exterior of the convex hull of Λ. More concretel...

Are braid groups CAT(0)?...