13 problems

A limit group is a finitely generated subgroup of a group obtained from a free group by finitely many extensions of centralizers, and it is one-ended when it has one end. For a gro…

Let $G$ be the group and $a$, $d$, and $P$ the element, rank, and set of primes occurring in conditions (a), (b), and (c) of the surrounding discussion. Baby Remeslennikov Conjectu…

Let $G$ be a finite group, let $H\leq G$ be a finite subgroup such that $\operatorname{Cir}_{\mathbb{Q}}^G(H)$ is non-empty, and let $g\in G$ satisfy $H\cap H^g=1$. Let…

Let $\{A_\lambda\}$ be a family of groups and let $\star_\lambda A_\lambda$ be their free product. For a fixed factor $A_\lambda$, let $w$ be an element of the free product that is…

Let $A$ be a nontrivial group, set $H=A\star\mathbb Z$, and let \savebox{\@brx}{m@th{langle}}% mathopen{copy@brxkern-0.5wd@brxusebox{@brx}} wsavebox{@brx}{\m@th{\rangle}…

Let $A$ be a group, let $H=A\star\mathbb Z$, and let $p:H\to\mathbb Z$ be the standard projection to the $\mathbb Z$ factor. For $w\in H$, write \savebox{\@brx}{m@th{langle}…

Let $A$ and $B$ be torsion-free groups, set $H=A\star B$, and let \savebox{\@brx}{m@th{langle}}% mathopen{copy@brxkern-0.5wd@brxusebox{@brx}} wsavebox{@brx}{\m@th{\rangle}…

Let $G$ and $H$ be linear sofic groups, let $G\ast H$ be their free product, and let $F_2$ denote the free group of rank $2$. Write $\mathcal{D}^{lin}_G(n)$ for the linear sofic pr…

Let $\mathcal{A}$ and $\mathcal{B}$ be automata, and let $\Sigma(\mathcal{A})$ and $\Sigma(\mathcal{B})$ be the automaton semigroups generated by their states. Their free product i…

Walker's conjecture. For any fixed chain $c$ in $F_n$, ${\rm scl}(c_{\bm{o}})$ is piecewise quasilinear in $1/o_i$: there are some $p\in\mathbb{Z}_+$ and a finite partition of…

Free-product no-isolated-points conjecture. The space $\operatorname{CO}(G*H)$ either has no isolated points or is finite. In particular, if it is infinite, then…

Let $S$ and $T$ be finite semigroups, and let $S\star T$ denote their semigroup free product. An automaton semigroup is a semigroup recognized by a finite automaton acting on finit…

Let $G_j^k = \langle a_j \rangle / \langle a_j^{o_{j,k}} \rangle = \mathbb Z/o_{j,k}\mathbb Z$ and let $G_k = *_j G_j^k$. Fix $\Gamma$ such that $\Gamma \in B_1^H(G_k)$ for all $k$…