Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1004.3588

On the Conjugacy Problem in Groups and its Variants

Michèle Feltz
View original source ↗Read source document ↗Export metadata ↓

Abstract

This thesis deals with the conjugacy problem in groups and its twisted variants. We analyze recent results by Bogopolski, Martino, Maslakova and Ventura on the twisted conjugacy problem in free groups and its implication for the conjugacy problem in free-by-cyclic groups and some further group extensions. We also consider the doubly-twisted conjugacy problem in free groups. Staecker has developed an algorithm for deciding doubly-twisted conjugacy relations in the case where the involved homomorphisms satisfy a certain remnant inequality. We show how a similar condition affects the equalizer subgroup and raise new questions regarding this subgroup. As an application we discuss the Shpilrain-Ushakov authentication scheme based on the doubly-twisted conjugacy search problem in matrix semigroups over truncated polynomials over finite fields. Part of this thesis is devoted to the implementation and testing of some of the previously mentioned concepts in the GAP programming language.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michèle Feltz. 2010-04-20. On the Conjugacy Problem in Groups and its Variants. https://arxiv.org/abs/1004.3588

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Finiteness conditions on skew braces and solutions of the Yang-Baxter equation

A finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation gives rise to a structure skew brace $B(X,r)$ that is a $λ_f$-skew brace, i.e. every element has finitely many $λ$-images, and whose additive group is $FC$. This motivates the study of finiteness conditions on skew braces. We first study the general class of $λ_f$ skew braces and the subclass where the additive group is $FC$, showing that these properties share a resemblance to finite conjugacy, having an analog of the $FC$-center and several analogous structural results. Furthermore, by passing through the structure skew brace of a solution, this property measures whether elements are contained in a finite decomposition factor, identifying a class of infinite solutions that may exhibit similar properties to finite ones. Finally, we show that for a sub skew brace where both groups have finite index, both indices need to coincide and that such a sub skew brace contains a strong left ideal of finite index.

math.GR↗
arXiv2026

Non-uniform exponential growth and the decay of growth rates in growing dimensions

We provide the first example of a finitely presented, and the first example of a simple, group of non-uniform exponential growth. The example is given by Thompson's group $V$. Our methods also show that the infimal exponential growth rates of $\mathrm{Aut}(F_{2^{n+2}})$ and of $\mathrm{EL}_{2^{n+2}}(R)$, for every finitely generated ring $R$, tend to $1$. As an application, we obtain the first example of an acylindrically hyperbolic group, and the first example of a Kazhdan group, of non-uniform exponential growth.

math.GR↗
arXiv2026

Solvable Supplements to Normalizers of Cyclic 2-Subgroups

Amberg and Kazarin proved that a finite group is solvable if the normalizer of every cyclic subgroup of prime power order has a solvable supplement. We substantially relax this hypothesis by requiring it only for cyclic $2$-subgroups. This condition, denoted by $\mathrm{SSN}_2$, sharply restricts the nonabelian composition factors of the group to the family $\PSL_2(q)$, where $q\geq7$ is a prime power satisfying $q\equiv3\pmod4$. Conversely, this family is precisely the nonabelian finite simple groups that satisfy $\mathrm{SSN}_2$. Consequently, a finite group satisfying $\mathrm{SSN}_2$ is solvable if and only if it has no section isomorphic to one of these groups.

math.GR↗