21 problems

Let ${\cal W}$ be an arbitrary element of the braid group $B_n$, and let $W$ be a word in the band generators representing it. The authors' algorithm also generalizes to the conjug…

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 $B_n^+$ be the positive braid monoid, let $n$ be fixed, and let $a\in B_n^+$ have word length $l$. Write $C^+(a)=C(a)\cap B_n^+$ for the positive conjugacy class of $a$. Thurst…

Polynomial-time conjugacy algorithm conjecture. There is an algorithmic solution to the conjugacy problem in $B_n$, using the combinatorial approach described in this paper, whose…

Let $G$ be a Garside group, let $x$ be a rigid element, and consider the conjugacy graph of $x^n$. Its vertices have levels, with level $1$ and maximal-level vertices as in the sou…

Let $G$ be a Garside group. For an element $y\in G$ that is conjugate to a rigid element and has a rigid power, let $r(y)$ be the smallest positive integer such that $y^{r(y)}$ is…

Let $G$ be a Garside group equipped with a Garside structure, and let $SC(x^n)$ denote the sliding circuits set of $x^n$. Uniform periodicity conjecture. There exists a finite set…

Let $G$ be a Garside group and let $x\in G$ be rigid. For each positive integer $n$, let $SC(x^n)$ denote the sliding circuits set of $x^n$. Boundedness conjecture. For any rigid e…

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 $G$ be the Baumslag-Gersten one-relator group … For a finitely generated group, the conjugator length function measures the smallest length of a conjugator between two conjugat…

Let $M$ be the Garside monoid under consideration, let $P$ be a standard parabolic submonoid of $M$, let $b\in P$ and $g\in M$ be such that $g$ is $P$-reduced and $b^g$ is defined.…

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…

Rips's conjecture. This should be true for all finitely presented groups.

Polynomial bound conjecture. There exists a constant $C$ such that, for every rigid braid $x$ with $N$ strands and Garside-length $L$,

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 $M$ be a positive homogeneously presented cancellative tame monoid whose idealistic subsemigroup is finitely generated. Solvability conjecture. The conjugacy problem in $M$ is…

Let $F_r$ be a finitely generated free group. A uniform polynomial-complexity algorithm for solving the conjugacy problem on $F_r$ is an algorithm whose polynomial complexity bound…

Let $x$ be a pseudo-Anosov braid with $n$ strings and canonical length $\ell$. Let $P_n$ be a polynomial depending on $n$. Polynomial growth conjecture. For all $n$, there exists a…

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…

Left–right ultra summit graph conjecture. For every element $x\in G$, not necessarily rigid, the left ultra summit set graph and the right ultra summit set graph of $x$ are isomorp…