Related papers: Twisted Conjugacy in Houghton's groups

Let $n\in \mathbb{N}$. Houghton's group $H_n$ is the group of permutations of $\{1,\dots, n\}\times \mathbb{N}$, that eventually act as a translation in each copy of $\mathbb{N}$. We prove the solvability of the conjugacy problem and…

We study twisted conjugacy classes of a family of groups which are called Houghton's groups $\mathcal{H}_n$ ($n \in\mathbb{N}$), the group of translations of $n$ rays of discrete points at infinity. We prove that the Houghton's groups…

We study the notion of twisted conjugacy separability (essentially introduced in our previous paper for a proof of twisted version of Burnside-Frobenius theorem) and some related properties. We give examples of groups with and without this…

Given a short exact sequence of groups with certain conditions, $1\to F\to G\to H\to 1$, we prove that $G$ has solvable conjugacy problem if and only if the corresponding action subgroup $A\leqslant Aut(F)$ is orbit decidable. From this, we…

For every finitely generated recursively presented group G we construct a finitely presented group H containing G such that G is (Frattini) embedded into H and the group H has solvable conjugacy problem if and only if G has solvable…

In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids $u,v\in B_n$ and an automorphism $\phi \in Aut (B_n)$, decides whether $v=(\phi (x))^{-1}ux$ for some $x\in B_n$. As…

There exist combable groups in which the conjugacy problem is unsolvable. The isomorphism problem is unsolvable for certain recursive sequences of finite presentations of combable groups.

Hougthon's groups H_n is a family of groups where each H_n consists of `translations at infinity' on n rays of discrete points emanating from the origin on the plane. Brown shows H_n has type FP_n-1 but not FP_n by constructing infinite…

We have proved in [Topology, 45 1 (2006)] that fundamental groups of oriented geometrizable 3-manifolds have a solvable conjugacy problem. We now consider the case of groups of non-oriented geometrizable 3-manifolds in order to conclude…

We discuss time complexity of The Conjugacy Problem in HNN-extensions of groups, in particular, in Miller's groups. We show that for "almost all", in some explicit sense, elements, the Conjugacy Problem is decidable in cubic time. It is…

In this paper we generalize techniques of Belk-Matucci to solve the conjugacy problem for every Thompson-like group $V_n(H)$, where $n \geq 2$ and $H$ is a subgroup of the symmetric group on $n$ elements. We use this to prove that, if $n…

This Thesis presents a 2-dimensional generalization of Houghtons' groups H_n. H_n is defined to be the group of all permutations p of a disjoint union of copies of the natural numbers N, with the property that each copy of N contains a…

In this paper we study the conjugacy problem in polycyclic groups. Our main result is that we construct polycyclic groups $G_n$ whose conjugacy problem is at least as hard as the subset sum problem with $n$ indeterminates. As such, the…

In this paper we provide an alternative solution to a result by Juh\'{a}sz that the twisted conjugacy problem for odd dihedral Artin groups is solvable, that is, groups with presentation $G(m) = \langle a,b \; | \; _{m}(a,b) = {}_{m}(b,a)…

The conjugacy problem belongs to algorithmic group theory. It is the following question: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zxz^{-1} =…

A group $G$ is twisted conjugacy separable if for every automorphism $\varphi$, distinct $\varphi$-twisted conjugacy classes can be separated in a finite quotient. Likewise, $G$ is completely twisted conjugacy separable if for any group $H$…

In 1951, Higman constructed a remarkable group $$H=\left\langle a,b,c,d \, \left| \, b^a = b^2, c^b = c^2, d^c = d^2, a^d = a^2 \right. \right\rangle$$ and used it to produce the first examples of infinite simple groups. By studying fixed…

We consider the following problem stated by Vdovin (2010) in the "Kourovka notebook" (Problem 17.41): Let $H$ be a solvable subgroup of a finite group $G$ that has no nontrivial solvable normal subgroups. Do there always exist five…

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…

A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We prove that the multiple conjugacy problem is solvable between two n-tuples A and B of…