Conjugacy separability and outer automorphism groups of certain HNN extensions

AbstractIn this note we first establish a criterion for the conjugacy separability of HNN extensions of polycyclic-by-finite groups with central associated subgroups. Using this result, we prove the equivalence of residually finiteness and conjugacy separability in these HNN extensions. Then we prove that the outer automorphism groups of these HNN extensions are residually finite if the HNN extensions themselves are residually finite

Hereditary conjugacy separability of right angled Artin groups and its applications

We prove that finite index subgroups of right angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group contains a conjugacy separable subgroup of finite index and has a residually finite outer automorphism group. Another consequence of the main result is that Bestvina-Brady groups are conjugacy separable and have solvable conjugacy proble

Quasi-Potency and Cyclic subgroup separability

We study the profinite topology on discrete groups and in particular the property of cyclic subgroup separability. We investigate the class of quasi-potent, cyclic subgroup separable groups, producing many examples and showing how it behaves with respect to certain group constructions

Decision problems and profinite completions of groups

We consider pairs of finitely presented, residually finite groups P\hookrightarrow\G for which the induced map of profinite completions \hat P\to \hat\G is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not $P$ is isomorphic to \G. We construct pairs for which the conjugacy problem in \G can be solved in quadratic time but the conjugacy problem in $P$ is unsolvable. Let $\mathcal J$ be the class of super-perfect groups that have a compact classifying space and no proper subgroups of finite index. We prove that there does not exist an algorithm that, given a finite presentation of a group \G and a guarantee that \G\in\mathcal J, can determine whether or not \G\cong\{1\}. We construct a finitely presented acyclic group \H and an integer $k$ such that there is no algorithm that can determine which $k$-generator subgroups of \H are perfect

Separability properties of higher-rank GBS groups

A rank $n$ generalized Baumslag-Solitar group is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to $\mathbb{Z}^n$. In this paper we classify these groups in terms of their separability properties. Specifically, we determine when they are residually finite, subgroup separable and cyclic subgroup separable