Mathematics Problem Archive

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...

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...

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...