Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 2, 183–362
Issue 1, 1–182

Volume 16, 5 issues

Volume 15, 5 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Author Index
Coming Soon
Other MSP Journals
The $334$-triangle graph of $\mathrm{SL}_3(\mathbb{Z})$

Eric S. Egge and Michaela A. Polley

Vol. 15 (2022), No. 3, 537–546

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.

chromatic number of a graph, generators and relations, special linear group, thin group, triangle group
Mathematical Subject Classification
Primary: 05C25
Received: 4 November 2021
Revised: 23 November 2021
Accepted: 25 November 2021
Published: 2 December 2022

Communicated by Stephan Garcia
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