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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 $\mathrm{\Delta }334$ triangle group in ${\mathrm{SL}}_{3}\left(ℤ\right)$ 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 $\mathrm{\Delta }334$ in the group. We provide examples of our graph for a variety of groups, and we use information about the graph for ${\mathrm{SL}}_{3}\left(ℤ∕2ℤ\right)$ to show that the chromatic number of the graph for ${\mathrm{SL}}_{3}\left(ℤ\right)$ is at most 8. By generating a portion of the graph for ${\mathrm{SL}}_{3}\left(ℤ\right)$ 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
Primary: 05C25