Transitions of the Diagonal Cartan Subgroup in SL(n,R)

A geometric transition is a continuous path of geometries which abruptly changes type in the limit. The most intuitive example is to imagine blowing up a sphere so that eventually it becomes so large, it looks like a plane. This is a transition from spherical geometry to Euclidean geometry.

We will study limits of the Cartan subgroup in $SL(n,R)$. A limit group is the limit under a sequence of conjugations of the Cartan subgroup in $SL(n,R)$. We will show using the hyperreal numbers that in $SL(3,R)$ there are 5 limit groups, each determined by a degenerate triangle.

In the second part of the talk, we will show that for $n \geq 7$, there are infinitely many nonconjugate limit groups of the Cartan subgroup.