论文标题
$ SL_3（{\ Mathbb Z}）$的334-Triangle图$
The 334-Triangle Graph of $SL_3({\mathbb Z})$
论文作者
论文摘要
Long，Reid和Thistlewaite表明，在$ SL_3（{\ Mathbb Z}）中$Δ334$ Triangle grout产生的某些组很薄，而其他人的状态未知。在本文中，我们采用了一种新方法：对于每个组，我们引入了一个新图，该图捕获了该组中$Δ334$的表示信息。我们为各个组提供图形的示例，并为$ sl_3（{\ Mathbb z}/2 {\ Mathbb z}）$使用有关图的信息，以表明$ sl_3（{\ Mathbb z}）的图形数量最多为八个。通过生成$ sl_3（{\ mathbb z}）$的图表的一部分，我们显示其色数至少为四个；我们猜想它等于四个。
Long, Reid, and Thistlewaite have shown that some groups generated by representations of the $Δ334$ 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 $Δ334$ in the group. We provide examples of our graph for a variety of groups, and we use information about the graph for $SL_3({\mathbb Z}/2{\mathbb Z})$ 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.
