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

TL;DR

This paper introduces a new graph-based method to analyze representations of the $ riangle 334$ triangle group in $SL_3({b Z})$, providing bounds on its chromatic number and insights into the group's structure.

Contribution

The paper develops a novel graph capturing representation information of $ riangle 334$ groups, applying it to $SL_3({b Z})$ to estimate its chromatic number.

Findings

Chromatic number of the graph for $SL_3({b Z})$ is at most eight.

Lower bound of the chromatic number is at least four.

Conjecture that the chromatic number equals four.

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.