19 problems

Bounded-region conjecture. The feature vectors of weights of the Whitehead graphs of elements from $F$ are separated into bounded regions in the corresponding space. Each such regi…

Nielsen prevalence conjecture.

Constructive membership testing difficulty conjecture. CMT is difficult whenever the group $G$ either involves a large abelian group as a quotient of a normal subgroup or has nonab…

Let $V_n$ be the Vershik group of rank $n$, and let $u\in V_n$ be a word with minimal generator length $l(u)$. Its Knuth–Bendix normal form is denoted by $\overline{u}$. Normal-for…

Let $F_n$ be a free group of rank $n$, let $u\in F_n$, and let $A(u)$ denote the set of automorphic images of $u$ having the same length as $u$. Assume that the length of $u$ is ir…

Let $B_n$ be the braid group on $n$ strands, and measure the input size by the word lengths of the elements involved in the conjugacy problem. Birman–Ko–Lee conjecture. For every f…

Let $G$ be the Baumslag-Gersten one-relator group … The conjugacy problem asks whether two words in the generators of $G$ represent conjugate elements. Non-polynomial conjugacy alg…

Let $H_{2n+1}(\mathbb{Z})$ denote the $(2n+1)$-dimensional Heisenberg group. Bodart's conjecture. Rational subset membership is decidable in every Heisenberg group…

Let $A$ be the right-angled Artin group for which Bridson constructed a finitely presented subgroup of $A\times A$ with undecidable conjugacy and membership problems. A right-angle…

Let $F$ be a finitely generated free group, let $Gamma=F\rtimes_{alpha}langle trangle$ be a unipotent polynomial suspension, and let $\operatorname{Aut}(Gamma,F)$ denote the automo…

For each integer $k\geq 2$, let … The subgroup membership conjecture for $B(P_{3,k})$. The subgroup membership problem is decidable in $B(P_{3,k})$ for every $k\geq 2$. The subgrou…

Let $G$ be a group hyperbolic relative to finitely generated subgroups $H_1,\ldots,H_N$, and suppose that the generalized conjugacy problem (GCP) in each parabolic subgroup $H_i$ c…

Conjecture that Bridson's group is not coherent. The right-angled Artin group $A$ in Bridson's example cannot be coherent.

Let $G$ be a group that splits as a finite graph of finitely generated free groups. An algorithm is said to certify that $G$ has infinite stature with respect to its vertex groups…

Polynomial-time conjugacy conjecture. The conjugacy problem can be solved in polynomial time for all hyperbolic elements of $H$.

Let $\BG$ be the Baumslag group and call an element of $\BG$ hyperbolic if it is hyperbolic with respect to the relevant HNN-extension structure. The conjugacy problem for hyperbol…

Let $\BG$ be the Baumslag group … The conjugacy problem asks, given two words representing elements of $\BG$, whether those elements are conjugate. Diekert–Myasnikov–Weiß's conject…

Let $P=\left\langle X\mid R\right\rangle$ be a finite presentation of a non-abelian group $\overline{P}$, and let $W(X)$ denote the words in the generators $X$. For words…

Let $G$ be a Garside group with Garside element $\Delta$ and canonical length parameter $k$; let $T$, $M$, and $R$ be the quantities defined in the preceding algorithmic complexity…