Conjugacy and period in symmetric diagram groups

Matteo Tarocchi

Home/Archive/arXiv

arXiv · 2610.05255

Conjugacy and period in symmetric diagram groups

Matteo Tarocchi

View original source ↗Read source document ↗Export metadata ↓

Abstract

We show how to solve the conjugacy (search) problem and the problem of determining whether a given element is periodic in symmetric diagram groups based on finite semigroup presentations.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Matteo Tarocchi. 2026-10-04. Conjugacy and period in symmetric diagram groups. https://arxiv.org/abs/2610.05255

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

KEEP EXPLORING

Related papers

arXiv2026

Unsolved Problems in Group Theory. The Kourovka Notebook

This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2--4 years since 1965. This is the 21st edition, which contains 150 new problems and a number of comments on problems from the previous editions.

math.GR↗

arXiv2026

On embeddings of the difference graph of the intersection power graph and the power graph

The power graph of a finite group $G$ is a simple undirected graph with vertex set $G$ and two vertices are adjacent if one is a power of the other. The intersection power graph of a finite group $G$ is a simple undirected graph with vertex set $G$ and two vertices $x$, $y$ are adjacent if $\langle x\rangle \cap \langle y \rangle \neq \{e\}$. The difference graph $\mathcal{D}(G)$ of a finite group $G$ is the difference of the intersection power graph $\mathcal{G}_{1}(G)$ and power graph $\mathcal{P}(G)$ with all isolated vertices removed. We characterized all the finite nilpotent groups $G$ such that the difference graph is planar. Further, we determine all the finite nilpotent groups whose difference graph has genus at most $2$. Moreover, we prove that there does not exist any group whose difference graph is projective planar.

math.GR↗

arXiv2026

Soficity of One-Relator Groups and Reducible Presentations

We prove that every one-relator group is sofic, answering a question of Nate Brown. More generally, every group admitting a reducible presentation without proper powers is sofic, with one proper-power relation allowed at the final reducible step. The proof starts from the endpoint-preserving edge replacements of Poulin--Wróbel. We retain the free-group word carried by each terminal replacement and use it to define an $F(S)$-valued cocycle repair; in its relative form, the repair preserves the previously chosen generator coordinates while controlling the new relator defect. Coinduction transports the lower cocycle through successive one-relator-product extensions, allowing the construction to be iterated along reducible presentations. For a final relator $w^m$, cyclic translates of the repaired set amplify a zero-defect set of measure close to $1/m$ to one of measure close to one. A cocycle criterion then converts arbitrarily small relator defect into finite permutation approximations via a treeable skew-product orbit relation.

math.GR↗