Mathematics Problem Archive

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

For a smooth strictly convex hypersurface $M\subset\mathbb{R}^{2n}$, find lower bounds for the number of $p$-periodic outer-billiard orbits for values...

Let $f(x,y)$ be a polynomial for which $0$ is a nonsingular value, let $\gamma$ be an oval component of $f(x,y)=0$, and assume that $\gamma_\varepsilo...

For the ellipsoid $x^2/a+y^2/b+z^2/c=1$ in Minkowski space with metric $dx^2+dy^2-dz^2$, find conditions on $a,b,c>0$ ensuring the existence of an $(n...

Assume the origami hyperbolic-paraboloid pattern has invisible straight folds along a chosen diagonal of each elementary trapezoid. What is the shape ...

Let $\gamma$ be a smooth arc agreeing to all orders with the $x$-axis at endpoints $(0,0)$ and $(1,0)$, and iterate $T(\gamma)=\gamma+\gamma'$. Unless...

Given an oriented oval $\gamma$, let $\gamma_1$ be the locus of endpoints of its oriented unit tangent segments, and iterate this construction. If eve...

Extend the known results on projectively self-dual polygons and curves to projectively self-dual polyhedra and to projectively self-dual polygons in m...

Find conceptual proofs and possible generalizations of the new projective-geometry configuration theorems described in the source. In particular, prov...

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

For a fixed combinatorial dissection of a square into $n$ triangles, the ordered triangle areas satisfy a polynomial relation depending only on the co...

Suppose $G$ admits a finite $K(G,1)$. If $G$ does not contain any Baumslag-Solitar subgroups $BS(m,n)$, is $G$ necessarily hyperbolic? If $G$ embeds i...

Suppose G admits a finite K(G, 1), does not contain Z × Z, and whenever x ∈ G is an infinite order element such that xm and xn are conjugate, then |m|...

(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...

(Mitra) Let XG be a finite 2-complex with fundamental group G. Let XH be a cover corresponding to the f.p. subgroup H. Let I(x) denote the injectivity...

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

(Whyte) Let Γ be a 1-ended hyperbolic group. Can a finite index subgroup of Γ be isomorphic to a subgroup of Γ of infinite index?...

(Swarup) Prove the combination theorem for relatively hyperbolic groups....

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 $n\to\infty$...

(Epstein) Let $G$ be a word-hyperbolic group and $\partial G$ its boundary. Is there an algorithm to compute $\check H^i(\partial G)\cong H^{i+1}(G,\m...

(M. Mitra) Let G be a word-hyperbolic group and H a word-hyperbolic subgroup. Does the inclusion H → G extend to a continuous map between the boundari...

(Swarup) Suppose $G$ is a hyperbolic group which is a graph of hyperbolic groups such that all edge-to-vertex inclusions are quasi-isometric embedding...

(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...

(Swarup) Is there a proof of Johannson’s theorem that Out(π1M) is virtually generated by Dehn twists for M a Haken 3-manifold along the lines of Rips-...

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

(Exercise in [BGS85, p.2]) Take a closed surface S of genus ≥2. Let V = S × S and let Σ ⊂V denote the diagonal. Let ˜V be a nontrivially ramified fini...

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...

(D. Wise) Let G act properly discontinuously and cocompactly on a CAT(0) space (or let G be automatic). Consider two elements a, b of G. Does there ex...

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

(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...

(Kim Ruane) Let $G$ be a Coxeter group, for example a right-angled Coxeter group, acting properly discontinuously by isometries on a CAT(0) space $X$....

(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?...

(Ross Geoghegan) A proper CAT(0) space $M$ is almost geodesically complete if there is $R\geq0$ such that for all $a,b\in M$ there is an infinite geod...

(Dani Wise) A triplane is a CAT(0) space obtained by gluing three Euclidean half-planes along their boundaries, and a CAT(0) space $X$ has isolated fl...

(Hanna Neumann Conjecture) If A and B are nontrivial subgroups of a free group, then rk(A ∩B) −1 ≤(rk(A) −1)(rk(B) −1)....

(Swarup) If G is a Fuchsian group, define area(G) to be the area of the convex core of H2/A. Since area(A) = 2π(rk(A)−1) for Fuchsian groups which are...

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

Conjecture. Limit groups are CAT(0)....

Characterize groups of the form Fm ∗Z Fn which are limit groups....

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

Conjecture: for $n>1$, the set $$\{(x_1,\ldots,x_n)\in F_n^n:x_1,\ldots,x_n\text{ is a basis of }F_n\}$$ is not definable....

Conjecture. The only definable subgroups of Fn are cyclic and the entire group....

For $x\in[F_n,F_n]$, define its genus as the least $g$ such that $x$ is a product of $g$ commutators. Let $f(g)$ be a uniform bound, independent of $x...