Mathematics Problem Archive

Consider two nested convex plane domains, and let $T_1,T_2$ be their billiard ball maps on the oriented lines intersecting both domains. If $T_1\circ ...

Are there smooth convex curves, other than ellipses, simultaneously admitting one-parameter families of $p$- and $q$-periodic billiard trajectories fo...

For billiards inside an oval in the Lorentz plane with metric $ds^2=dx^2-dy^2$, is there an analogue of Birkhoff's theorem, with separate existence st...

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....

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|...

(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...

(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...

(Canary) Let $G$ be word-hyperbolic and $H$ a finitely presented subgroup of $G$. Suppose that for every $g\in G$ there is $n>0$ such that $g^n\in H$....

(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...

(Thurston) Is every closed hyperbolic 3-manifold finitely covered by one that fibers over the circle?...

(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?...

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

(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?...

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...

(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...