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arXiv · 2201.08908

The 334-Triangle Graph of $SL_3({\mathbb Z})$

Eric S. Egge·Michaela A. Polley
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Abstract

Long, Reid, and Thistlewaite have shown that some groups generated by representations of the $\Delta 334$ triangle group in $SL_3({\mathbb Z})$ are thin, while the status of others is unknown. In this paper we take a new approach: for each group we introduce a new graph that captures information about representations of $\Delta 334$ in the group. We provide examples of our graph for a variety of groups, and we use information about the graph for $SL_3({\mathbb Z}/2{\mathbb Z})$ to show that the chromatic number of the graph for $SL_3({\mathbb Z})$ is at most eight. By generating a portion of the graph for $SL_3({\mathbb Z})$ we show its chromatic number is at least four; we conjecture it is equal to four.

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Eric S. Egge, Michaela A. Polley. 2022-01-21. The 334-Triangle Graph of $SL_3({\mathbb Z})$. https://arxiv.org/abs/2201.08908

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