Mathematics Problem Archive

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...

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?...

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)?...

Extend Rips’ theory to higher-dimensional buildings, e.g. products ofR-trees. Rank rigidity. Let X be a CAT (0) metric space. The space X is said to b...

Suppose that Y is a compact finitedimensional locally CAT (0) metric space of rank n ≥ 2. Then either the universal cover of Y splits (nontrivially) as...

Under the above assumptions, is it true that Y has coarsely trivial πm for m ≥ 2?...

Does the Coarse Whitehead Conjecture hold if G is hyperbolic?...

Is there a useful list of surgery procedures which generates all smooth four-manifolds of a given homotopy type?...

Find a structure or conjectural decomposition for smooth four-manifolds that could play the guiding role that Thurston's Geometrization Conjecture pla...

Let $f,g$ be smooth vector fields on an $n$-dimensional manifold $M$, and consider $\dot q=f(q)+ug(q)$, $|u|\leq1$, with fixed endpoint. For a generic...

Let $\gamma_i:[0,1]\to M$, $i=0,1$, be smooth admissible paths of a sub-Riemannian structure with $\gamma_0(0)=\gamma_1(0)=q_0$ and $\dot\gamma_0(0)\w...

For the endpoint map from the $H^1$ Hilbert manifold of admissible paths starting at $q_0$ to $M$, can the singular curves starting at $q_0$ fill all ...

Find a $C^1$-classification of the germs of sub-Riemannian spheres at points of optimal singular curves for generic metrics. In particular, obtain suc...

A distribution is singular transitive if any two points can be connected by a concatenation of singular curves. Does singular transitivity imply that ...

There exist values of $\lambda_1$ and $\lambda_2$ such that the spectrum $\sigma(H)$ of $H$ is a Cantorval; that is, the spectrum is the closure of it...

Determine the position of sharp rings in the diffraction measure of a Pinwheel Tiling and their intensity....

Does the diffraction measure of the Pinwheel Tiling contain an absolutely continuous component?...

Let $p(n)$ count radius-$n$ patches in an aperiodic repetitive tiling of dimension $d$, and let $\Omega$ be its tiling space. If $p(n)=O(n^d)$, must t...

Classify tilings having a geometric property such as bounded-displacement equivalence (BD), bi-Lipschitz equivalence (BL), or linear repetitivity (LR)...

Develop and study new geometric properties, analogous but not identical to BD, BL, etc., that are invariant under MLD, topological conjugacy, or homeo...

But deeper structures (if they exist at all) that lie at the roots of Dirac operators and of minimal hypersurfaces are yet to be revealed....

Identify the most general classes of geometric objects having properties analogous to those of $C^2$ Riemannian manifolds with $\operatorname{Sc}\geq\...