Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2412.02892

On the biautomaticity of CAT(0) triangle-square groups

Mateusz Kandybo
View original source ↗Read source document ↗Export metadata ↓

Abstract

Following the research from the paper "Triangles, squares and geodesics" (arXiv:0910.5688) of Rena Levitt and Jon McCammond we investigate the properties of groups acting on CAT(0) triangle-square complexes, focusing mostly on biautomaticity of such groups. In particular we show two examples of nonpositively curved triangle-square complexes $X_1$ and $X_2$, such that their universal covers violate conjectures given in the aforementioned paper. This shows that the Gersten-Short geodesics cannot be used as a way of proving biautomaticity of groups acting on such complexes. Lastly we give a proof of biautomaticity of $\pi_1(X_1)$, however the biautomaticity of $\pi_1(X_2)$ remains unknown.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mateusz Kandybo. 2024-12-03. On the biautomaticity of CAT(0) triangle-square groups. https://arxiv.org/abs/2412.02892

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

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗
arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗
arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗