14 problems

Torsion-free hyperbolic twisted-conjugacy conjecture. The result for free groups should generalize to torsion-free hyperbolic groups and their automorphisms; in particular, the cor…

Let $\Gamma$ be a group with exponential growth, and let $\phi:\Gamma\rightarrow\Gamma$ be an injective endomorphism. Two elements of $\Gamma$ are $\phi$-conjugate if there exists…

Let $G$ be a countable discrete group and let $phi:G\to G$ be an endomorphism. Let $R(\phi)$ be the Reidemeister number, and let $R_*(\phi)$ be the dimension of the space of twiste…

Injective endomorphism conjecture. If $\phi$ is injective and $G$ has exponential growth, then

Property $R_\infty$ conjecture. Every finitely generated residually finite group that is not solvable-by-finite has property $R_\infty$. In the weaker version, every finitely gener…

Broad exponential-growth vanishing conjecture. The degree of twisted commutativity and the twisted conjugacy ratio are equal to zero in these classes of groups.

Let $\Gamma$ be a virtually polycyclic group and let $\varphi:\Gamma\to\Gamma$ be an endomorphism. When the Reidemeister zeta function is well-defined, it is given by … where…

Let $\Gamma$ be a finitely generated residually finite group. An automorphism $\varphi$ of $\Gamma$ has infinite Reidemeister number when $R(\varphi)=\infty$; a group is $R_\infty$…

Let $\mathbf{B}$ be a Borel subgroup of a split, connected, reductive, non-commutative linear algebraic group $\mathbf{G}$, all defined over a global field. An $S$-arithmetic subgr…

Let $\phi$ be an endomorphism of a group $\Gamma$, and let $R(\phi)$ denote its Reidemeister number, namely the number of $\phi$-twisted conjugacy classes. Write…

Twisted conjugacy separability domination conjecture. For every automorphism $\varphi\in\operatorname{Aut}(N)$,

Let $G$ be a group, let $\operatorname{Aut}(G)$ denote its automorphism group, and for $\varphi\in\operatorname{Aut}(G)$ let $[e]_{\varphi}$ be the twisted conjugacy class of the i…

Centerless-group inner-automorphism conjecture. If $G$ has no center, then there exists an inner automorphism $\varphi$ such that $[e]_{\varphi}$ is not a subgroup of $G$.

Twisted-conjugacy subgroup conjecture. If $[e]_{cvarphi}$ is a subgroup of $G$ for every automorphism $cvarphi eq$ in $cmathrm{Aut}G$, then $G$ is nilpotent. If $G$ is also finitel…