Conjugacy Limits of the Cartan Subgroup in SL(3,R)

TL;DR

This paper classifies the possible limit groups of conjugates of the Cartan subgroup in SL(3,R), providing a detailed criterion for convergence and describing five distinct limit groups.

Contribution

It identifies and characterizes five possible limit groups of conjugates of the Cartan subgroup in SL(3,R), linking them to nonstandard triangles.

Findings

Five possible limit groups identified and classified.

A criterion for convergence of conjugates to each limit group.

Connection between limit groups and equivalence classes of nonstandard triangles.

Abstract

A limit group is the limit of a sequence of conjugates of the diagonal Cartan subgroup, C, of SL(3,R). We show C has 5 possible limit groups, up to conjugacy. Each limit group is determined by an equivalence class of nonstandard triangle, and we give a criterion for a sequence of conjugates of C to converge to each of the 5 limit groups.