Mathematics Problem Archive

Does every polygonal outer billiard in the hyperbolic plane have periodic orbits, possibly lying on the circle at infinity?...

Describe the polygonal outer billiard tables in the hyperbolic plane for which every orbit is periodic....

Does there exist a totally skew embedded $3$-disc in $\mathbb{R}^7$?...

(Davis) If $G$ is word-hyperbolic, does the Rips complex $P_d(G)$ have an equivariant negatively curved metric for $d$ sufficiently large?...

(Gromov) Does every one-ended word-hyperbolic group contain a closed hyperbolic surface subgroup?...

(Gromov) For a given n is there an example of a hyperbolic group of dimension n in which every infinite index subgroup is free? Or in which there are ...

(Swarup) Suppose $H$ is a finitely presented subgroup of a word-hyperbolic group $G$ and has finite index in its normalizer. Assume that there is $n>0...

(Whyte) Let Γ be a 1-ended hyperbolic group which is not virtually a surface group. Can every infinite index subgroup be free?...

Is every word-hyperbolic group residually finite?...

(Dani Wise) Let $G^n$ denote the Cartesian product of $n$ copies of $G$, and let $\operatorname{rank}(G^n)$ be its smallest number of generators. If $...

(Dani Wise) Find 'nice' groups, for example CAT(0) or automatic groups, for which $\operatorname{rank}(G^n)$ does not tend to infinity as $n\to\infty$...

(Ian Leary) Is there a version of the Kan-Thurston theorem using only CAT(-1) groups, or word hyperbolic groups? (The statement should be: for any fin...

(Gromov) If $G$ admits a finite-dimensional $K(G,1)$, does $G$ act properly discontinuously by isometries on a complete CAT(0) space?...

(Eilenberg-Ganea) Is there a group G of cohomological dimension 2 and geometric dimension 3?...

Suppose a group G acts properly discontinuously and cocompactly by isometries on two CAT(0) spaces X and Y . Croke-Kleiner have examples where the bou...

Do CAT(0) (or (bi)automatic) groups satisfy the Tits alternative?...

Does every Artin group have a finite $K(G,1)$?...

(Eric Swenson) Let $X$ be a proper CAT(0) metric space and $G$ a finitely generated group acting properly discontinuously by isometries on $X$. (1) Ca...

(Ruth Charney) Classify Coxeter groups up to isomorphism....

(Ruth Charney) Classify Artin groups up to isomorphism....

(Ruth Charney) Are all [finite type] Artin groups CAT(0)?...

(Ruth Charney) Are all Artin groups automatic?...

(J. Cornick) If G is f.g. and the homological dimension hd G = 1, is G free?...

Characterize 1-relator groups which are limit groups....

(Swarup) Suppose G is a 1-ended finitely presented group that acts on a compact connected metric space X as a convergence group. What can be said abou...

(Jim Anderson) If G is a group of isometries of Hn, denote by Ax(G) the set of axes of the elements of G. If G1 and G2 are finitely generated and disc...

(Misha Kapovich) Is there a finitely-generated discrete subgroup of SO(n, 1) whose action on the limit set is not ergodic? Is not recurrent?...

(Ian Leary) Is there a group of finite vcd that does not act with finite stabilizers on an acyclic complex of dimension equal to its vcd?...

(Peter Kropholler) If G is FP over the rationals, is there a bound on the orders of finite subgroups of G?...

(Sageev) Is there a f.p. 1-ended group G with G ∼= G∗Z?...

(Papasoglou) Is there a f.p. torsion-free group G that does not split over a virtually abelian subgroup, but has infinitely many splittings over F2?...

Can Zp∞be embedded in an FP∞-group? Or in an FP3-group? Can an Fn-group be embedded in an Fn+1-group (n ≥2)?...

Is there a finitely presented group $G=F/N$, with $F$ a finite-rank free group, such that $$d_G\!\left(N/[N,N]\right)<d_F(N)<\infty?$$ Here $d_F(N)$ i...

(Andrews-Curtis) If K and L are simple homotopy equivalent finite 2-complexes, can one transform K to L by a sequence of elementary collapses and expa...

(Wise) Is there a finite aspherical 2-complex X with π1(X) coherent and with χ(X) ≥2?...

(Igor Belegredek) Let X be a non-positively curved symmetric space. Find conditions on a group Γ so that the space of conjugacy classes of faithful di...

(Belegredek) Is there a 3-complex X (not necessarily aspherical) which is not homotopy equivalent to a 2-complex but H³(X; {G}) = 0 for all local coef...

Study the quasi-isometry group QI(Rn). How big is it?...

(Kleiner) What are the q.i’s of the Gromov-Thurston examples of negatively pinched manifolds?...

(Feighn) Let $\phi:F_n\to F_n$ be an automorphism and let $M_\phi$ be its mapping torus. Classify these groups up to quasi-isometry....

(Bridson) Is G×Z quasi-isometric to G for G =Thompson’s group? Any f.g. group?...

(Lubotzky) Does Out(Fn) have the congruence subgroup property?...

Does Out(Fn) (n > 2) have a right orderable subgroup of finite index?...

Do there exist f.g. free subgroups of MCG(S) consisting of identity and pseudo-Anosov mapping classes which are not Schottky?...

For $\operatorname{Out}(F_n)$, do there exist finitely generated free subgroups consisting of the identity and irreducible automorphisms which are not...

For $\operatorname{Out}(F_n)$, is the natural map $\operatorname{Out}(F_n)\to QI(\operatorname{Out}(F_n))$ an isomorphism? Does every quasi-isometry o...

Is MCG(Sg,b,n) linear?...

(Yves de Cornulier) Let G be residually torsion-free nilpotent. Is G Haagerup (= a-(T)-menable)?...

(Kevin Whyte) Let K be a finite complex with π = π(K) amenable. Is there a uniform bound to the betti numbers of finite covers of K?...

(Henry Glover) Does for every finite graph G the following 1−2−∞- conjecture hold: G is planar, a double cover of G is planar, or no finite cover is p...