Limits of the Diagonal Cartan Subgroup in SL(n,R) and SL(n, Q_p)

A conjugacy limit group is the limit of a sequence of conjugates of the positive diagonal Cartan

subgroup, C \leq SL(n) in the Chabauty topology.   Over R, the group C is naturally associated to a projective n-1 simplex.  We can compute the conjugacy limits of C by collapsing the n-1 simplex in different ways.  In low dimensions, we enumerate all possible ways of doing this.  In higher dimensions we show there are infinitely many non-conjugate limits of C. 

In the Q_p case, SL(n,Q_p) has an associated p+1 regular affine building.  (We'll give a gentle introduction to buildings in the talk).  The group C stabilizes and apartment in this building, and limits are contained in the parabolic subgroups stabilizing the facets in the spherical building at infinity. There is a strong interplay between the conjugacy limit groups and the geometry of the building, which we exploit to extend some of the results above.  The Q_p part is joint work with Corina Ciobotaru and Alain Valette.