Limits Under Conjugacy of the Diagonal Cartan Subgroup in SL(n,R)
A conjugacy limit group is the limit of a sequence of conjugates of the positive diagonal Cartan subgroup, C ≤ SL(3,R). In chapter 6, we prove a variant of a theorem of Haettel, and show that up to conjugacy in SL(3,R), the positive diagonal Cartan subgroup has 5 possible conjugacy limit groups. Each conjugacy limit group is determined by a nonstandard triangle. We give a criterion for a sequence of conjugates of C to converge to each of the 5 conjugacy limit groups.
In chapter 8, we give a quadratic lower bound on the dimension of the space of conjugacy classes of subgroups of SL(n,R) that are limits under conjugacy of the positive diagonal subgroup. We give the first explicit examples of abelian (n − 1)-dimensional subgroups of SL(n,R) which are not such a limit, however all such abelian groups are limits of the positive diagonal group iff n ≤ 4.
In chapter 4, we classify all subgroups of PGL(4,R) isomorphic to (R^3,+), up to conjugacy, and Haettel shows each is a limit of the positive diagonal Cartan subgroup. By taking subgroups of these groups satisfying certain properties, we show there are 4 possible families of generalized cusps up to projective equivalence in dimension 3, and describe each cusp.