44 problems

Let $\mathcal{F}=\prod_{\alpha\in I}^*G_\alpha$ be a free product of groups, and let $q^*$ be the minimum of the orders greater than $2$ of finite subgroups of the factors, with…

Shapiro's conjecture. A finitely generated group is geodetic if and only if it is plain.

Holt–Rees–Röver–Thomas conjecture. The class $co\mathcal{CF}$ is not closed under taking free products; specifically,

Let … be a reduced free product of unital $C^*$-algebras with faithful states $\phi_1$ and $\phi_2$. Pure infiniteness conjecture. Whenever $A$ is simple and at least one of…

Let $G_{1},G_{2},\ldots,G_{n}$ be finite groups, let $s=\max_{i}d(G_{i})$, and let … be their free product. Write $\widehat{\Gamma}$ for the profinite completion of $\Gamma$, and l…

Let $G=\prod^*_{\lambda\in\Lambda}G_\lambda$ and let $H$ be a subgroup of $G$. In the proofs of Higgins's theorem and Kuroš's theorem, let $\tau$ denote the tree used to obtain the…

Pointed-tree space conjecture. The space $L(G)$ is an $\underline{E}\Sigma Aut(G)$-space in addition to being an $\underline{E}\Sigma Aut_1(G)$-space.

Let $A$ and $B$ be C$^*$-algebras equipped with faithful tracial states, and let $A*B$ denote their reduced free product with respect to these states. The stable-rank-one conjectur…

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 $A$ be a nontrivial group, let $G=A\star\mathbb Z$, and let $p:G\to\mathbb Z$ be the natural surjection. For $w\in G$, write $\langle\!\langle w\rangle\!\rangle$ for the normal…

Let $I$ be an index set with $|I|\geq 3$, and let $G=\star_{i\in I}A_i$ be a free product of nontrivial groups $A_i$. Gordon's conjecture. The group $G$ has normal rank greater tha…

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…

Madlener–Otto conjecture. A group admits a finite confluent length-reducing rewriting system if and only if it is plain.

Let $m\in\mathbb{N}$ and let … For $\gamma\in\mathcal{F}$, define … and let $\operatorname{Crit}(\gamma)$ be the subgroups achieving this maximum. Puder–Zimhony's conjecture. If…

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 $H=\star_{\lambda\in\Lambda}G_\lambda$ be a free product of nontrivial groups, with $|\Lambda|\ge 2$, and let \savebox{\@brx}{m@th{langle}…

Let $G$ be a nontrivial group. Adding one generator and then one relator means forming a quotient of $G\star\mathbb Z$ by the normal closure of an element. Kervaire's conjecture. T…

Amalgam conjecture. Every finitely generated group that does not have property FA is a nontrivial free product with amalgamation.

Assume that $Gamma$ is a group satisfying the paper's Assumption on $Gamma$, and let $gamma\text{ be a non-torsion element of }Gamma$. Define … Let $\mathrm{Crit}(\gamma)$ be the n…

For $n\geq 2$, let $R_n,U_n,T_n\in\operatorname{GL}_n(\mathbb{C})$ be the matrices defined above, let $H_n=\langle R_n,U_n,T_n\rangle$, and write $\langle R_n\rangle$ and…

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…