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Let ${cal C}$ be a nonempty class of finite groups closed under taking subgroups, homomorphic images and extensions. A subgroup $H$ of an abstract residually ${cal C}$ group $R$ is said to be conjugacy ${cal C}$-distinguished if whenever $yin R$, then $y$ has a conjugate in $H$ if and only if the same holds for the images of $y$ and $H$ in every quotient group $R/Nin {cal C}$ of $R$. We prove that in a group having a normal free subgroup $Phi$ such that $R/Phi$ is in ${cal C}$, every finitely generated subgroup is conjugacy ${cal C}$-distinguished. We also prove that finitely generated subgroups of limit groups, of Lyndon groups and certain one-relator groups are conjugacy distinguished (${cal C}$ here is the class of all finite groups).
We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundam
In this paper, we consider the conjugacy growth function of a group, which counts the number of conjugacy classes which intersect a ball of radius $n$ centered at the identity. We prove that in the case of virtually polycyclic groups, this function i
In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is $0$ for all groups except
We give a unified solution to the conjugacy problem for Thompsons groups F, T, and V. The solution uses strand diagrams, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompsons groups. Strand diagrams are clo
We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of