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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
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Abstract |
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Long, Reid, and Thistlethwaite have shown that some groups generated by representations of the triangle group in 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 in the group. We provide examples of our graph for a variety of groups, and we use information about the graph for to show that the chromatic number of the graph for is at most 8. By generating a portion of the graph for we show its chromatic number is at least 4; we conjecture it is equal to 4. |
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Keywords
chromatic number of a graph, generators and relations,
special linear group, thin group, triangle group
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Mathematical Subject Classification
Primary: 05C25
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Milestones
Received: 4 November 2021
Revised: 23 November 2021
Accepted: 25 November 2021
Published: 2 December 2022
Communicated by Stephan Garcia
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Authors |
| Eric S. Egge | |
| Department of Mathematics and
Statistics Carleton College Northfield, MN United States |
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| Michaela A. Polley | |
| Department of Mathematics and
Statistics Carleton College Northfield, MN United States |
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