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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
Abstract

Long, Reid, and Thistlethwaite have shown that some groups generated by representations of the Δ334 triangle group in SL 3() 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 Δ334 in the group. We provide examples of our graph for a variety of groups, and we use information about the graph for SL 3(2) to show that the chromatic number of the graph for SL 3() is at most 8. By generating a portion of the graph for SL 3() 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