Mathematics Problem Archive

(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 $g^n\in H$....

(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...

(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 $X$....

(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...

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

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...

Is there a proper closed subgroup of Homeo+(S1) that acts transitively on (unordered) 4-tuples? Or k-tuples (k ≠ 3)? Relation to earthquakes?...

(Misha Kapovich) Suppose $G$ is a finitely generated Kleinian group in $\operatorname{Isom}(\mathbb{H}^n)$. Is $$\delta(G)\leq\operatorname{vcd}(G),$$...

(Swarup) Starting with a finitely presented group $G$, take maximal graph-of-groups decompositions alternately over finite and over two-ended edge gro...

(Shalen) If a finitely presented group G acts nontrivially (i.e. without global fixed points) on an R-tree, does it act nontrivially on a simplicial t...

(Mohan Ramachandran) For a finitely presented group $G$, consider: (A) some finite-index subgroup of $G$ admits a nontrivial action on a simplicial tr...

Compute the asymptotic dimension of CAT(0) groups, Out(Fn), mapping class groups, nonuniform lattices, Thompson’s group. Is there a group of finite ty...

Suppose $H=F/N$ is finitely presented and contains $C^n$ for every $n$. Write $G_n=H*_{C^n}H=F*F/N_n$. If $C$ is nontrivial and finite, is $$\lim_{n\t...

(Bridson) Classify the mapping tori of automorphisms $\mathbb{Z}^n\to\mathbb{Z}^n$ up to quasi-isometry....

(Kapovich) (a) For closed orientable surfaces $\Sigma_g$ and $\Sigma_h$, is there a faithful representation $\pi_1(\Sigma_g)\to\operatorname{MCG}(\Sig...

Assuming that the fixed subgroup Fix(α) is cyclic, find a bound on the length of a generator of Fix(α) in terms of the complexity of α....

Which α preserve an order (invariant under right translations) on Fn? If α has periodic elements it cannot preserve an order. Are there other obstruct...

Can the mapping class group (or a finite index subgroup of it) of a closed surface be embedded into Out(Fn)? Into the mapping class group of a punctur...

(Grigorchuk) Aut(Fn) acts on the space ∂3Fn of triples of distinct ends of Fn. Denote by Yn the compact space (Cantor set) the quotient space of ∂3Fn ...

(Seymour Bachmut) Is SL2(K) finitely generated for K = Z[X, X−1] or K = F[X, X−1, Y, Y −1] for a field F?...

Can the odometer acting on the rooted binary tree be embedded in a nonabelian free pro-$2$ group?...

Determine the asymptotic length of a shortest nontrivial group law for the $n$-fold iterated wreath product of $C_2$....

Does every countable group have fixed price; that is, do all its free probability-measure-preserving actions have the same cost?...

Let $\Gamma$ act on $X$ by probability-measure-preserving maps and let $H\leq\Gamma$ have finite index. Is $\operatorname{cost}(H,X)-1=(\operatorname{...

Are two independent random subsets of $\mathbb{Z}$ almost surely quasi-isometric as metric spaces? What is the answer for other Cayley graphs, such as...

Let $P$ be a finite $p$-group of order $n$ and let $w$ be any word. Is the probability that $w$ is satisfied in $P$ at least $1/n$?...

Determine the asymptotic length of the shortest nontrivial word that is a law in every group of order $2^n$....

Which Kleinian groups admit closed quasi-Fuchsian surface subgroups?...

The CAT(0) cubical dimension of a group $G$ is the least dimension of a CAT(0) cubical space on which $G$ acts geometrically. Is there a sequence of c...

Let $\Gamma$ be a discrete group acting properly discontinuously by affine transformations on $\mathbb{R}^5$. Is $\Gamma$ virtually an extension of a ...

Classify all properly discontinuous affine actions of a given closed surface group on $\mathbb{R}^6$....

For a given right-angled Coxeter group, what is the least $n$ for which it admits a proper affine action on $\mathbb{R}^n$?...

Let $\Gamma$ be a Kleinian group of finite covolume, and let $\mathcal{C}(\Gamma)$ be the set of isomorphism classes of its finite quotient groups. Do...

For $\Gamma=F_r$ with $r\geq2$, does $\mathcal{C}(F_r)$ determine $F_r$ up to isomorphism?...

Let $\Gamma$ be a Kleinian group of finite covolume. Does its set $\mathcal{C}(\Gamma)$ of finite quotient isomorphism classes determine $\Gamma$ amon...

If $G$ is a Gromov-hyperbolic group whose boundary is homeomorphic to the limit set of a convex-cocompact Kleinian group, is $G$ virtually a convex-co...

For which Kleinian groups does the limit set contain a Sierpiński carpet? For which Kleinian groups does the limit set contain a continuum?...