435 problems

Let $G$ be a finitely generated group. Recall that $G$ is virtually free if it has a finite-index free subgroup. Ballier and Stein's conjecture. The group $G$ has decidable domino…

Let $G$ be a CK-admissible group acting geometrically on a Hadamard space $Y$. Let $T_{\infty}$ be the Bass–Serre tree associated to the graph-of-groups structure of $G$, and let…

Lytchak's Regular Point Conjecture. The Tits boundary $\partial_T X$ contains a regular point.

Let $J_n$ be the cactus group on $n$ strands, and let $k$ be a nonnegative integer. A free abelian subgroup of rank $k$ is a subgroup isomorphic to $\mathbb{Z}^k$. Free abelian sub…

Let $L$ be a flag complex, let $\mathbb{F}$ be a skew field, and let $\varphi\colon A_L\to\mathbb{Z}$ be an epimorphism. Suppose that $BB_L^\varphi$ is of type $\mathsf{F}$. Minima…

Let $M$ be a closed aspherical $3$-manifold and let $\Gamma=\pi_1(M)$. The paper's statements numbered through concern properties of $M$ and $\Gamma$ introduced earlier in the pape…

Commensurator discreteness conjecture. The group $\operatorname{Comm}_{\mathrm{PSL}_2(\mathbb{R})}(\Phi(H))$ is discrete provided that $H\neq\Phi(H)$. More generally,

Non-zero asymptotic excess conjecture. If $\Sigma_\Psi$ has non-zero asymptotic excess, then, for every admissible $\mathbf y$, the transformation $T_{\Psi,\mathbf y}$ either is no…

A polygonal linkage is a closed chain of segments with fixed edge lengths, and the swap group $SW_n$ is generated by the elementary swaps $s_i$ acting on its configurations. Let…

Let $G$ be a hyperbolic group whose boundary $\partial G$ is homeomorphic to $S^{n-1}$. An aspherical closed manifold is a closed manifold whose universal cover is contractible. Gr…

Let ${\cal M}_g$ be the mapping class group and let $H_b^2({\cal M}_g)$ denote its second bounded cohomology. Bounded-cohomology conjecture. For every genus $g$, the comparison map…

Let $R$ be an integral domain and let $G$ be a torsionfree group. An idempotent in the group ring $RG$ is an element $e$ satisfying $e^2=e$. Kaplansky's conjecture. Every idempoten…

Standard-action conjecture. Every quasi-action by $\Gamma$ on a Gromov hyperbolic metric space either has an invariant horoball or is standard.

Finiteness conjecture. If $\Gamma$ is an irreducible lattice in a higher rank Lie group (or in a nontrivial product of locally compact groups) $G$, there are finitely many Gromov h…

Let $S$ be a surface, let $\operatorname{Mod}(S)$ be its mapping class group, and let $\Lambda_G$ denote the limit set of a subgroup $G\leq \operatorname{Mod}(S)$ in the projective…

Let $X=G/K$ be an irreducible symmetric space of rank at least two, let $F$ be a maximal flat in $X$, and let $p_0\in F$. A geodesic in $F$ through $p_0$ is singular if it has the…

Let $\Gamma$ be a Kleinian group in $\operatorname{Mob}(\mathbb S^n)$ without parabolic elements, let $cd(\Gamma)$ denote its cohomological dimension, and let $\delta(\Gamma)$ deno…

Let $\Gamma$ be a lattice in a higher-rank Lie group, and let its absolute rank gradient be the infimum of $(d(H)-1)/[\Gamma:H]$ over finite-index subgroups $H\leq\Gamma$, where…

Let $(G_{1,1},X)$ be the group–space pair whose patterns are being studied, and let a pattern be a sequence encoded by parenthesized blocks; within a block, $(0,-1)$ or $(1,0)$ den…

Bounded packing conjecture for virtually polycyclic groups. Each subgroup of $P$ has bounded packing in $P$.

Let $\Sigma_{1}$ and $\Sigma_{2}$ be compact, orientable surfaces, and let $\mathcal{P}(\Sigma)$ denote the pants graph of a surface $\Sigma$. A subgraph is totally geodesic if eve…

Mühlherr's conjecture. There exists a reflection-preserving automorphism $\alpha$ of $W$ such that $\alpha(S)$ is sharp-angled with respect to $S'$.

Brady–McCammond–Mühlherr–Neumann conjecture. If

For a fixed finite generating set of ${\rm Mod}_g$, define the density of a subset $S$ by … Let $\Tor_g$ denote the set of finite-order elements of ${\rm Mod}_g$. Torsion-density c…

Gersten's conjecture. There is a recursive function $f\colon\mathbb{N}\to\mathbb{N}$, such that every finite presentation of a linear group has