The 334-triangle graph of SL3(ℤ)

Egge, Eric S.; Polley, Michaela A.

doi:10.2140/involve.2022.15.537

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The $334$-triangle graph of $\mathrm{SL}_3(\mathbb{Z})$

Eric S. Egge and Michaela A. Polley

Vol. 15 (2022), No. 3, 537–546

DOI: 10.2140/involve.2022.15.537

Abstract

Long, Reid, and Thistlethwaite have shown that some groups generated by representations of the $Δ 334$ triangle group in ${SL}_{3} (ℤ)$ are thin, while the status of others is unknown. We take a new approach: For each group we introduce a new graph that captures information about representations of $Δ 334$ in the group. We provide examples of our graph for a variety of groups, and we use information about the graph for ${SL}_{3} (ℤ ∕ 2 ℤ)$ to show that the chromatic number of the graph for ${SL}_{3} (ℤ)$ is at most 8. By generating a portion of the graph for ${SL}_{3} (ℤ)$ we show its chromatic number is at least 4; we conjecture it is equal to 4.

Keywords

chromatic number of a graph, generators and relations, special linear group, thin group, triangle group

Mathematical Subject Classification

Primary: 05C25

Milestones

Received: 4 November 2021

Revised: 23 November 2021

Accepted: 25 November 2021

Published: 2 December 2022

Communicated by Stephan Garcia

Authors

Eric S. Egge
Department of Mathematics and Statistics Carleton College Northfield, MN United States

Michaela A. Polley
Department of Mathematics and Statistics Carleton College Northfield, MN United States

Vol. 15, No. 3, 2022