Ross Geoghegan (Binghamton University - State University of NewYork)

"Horospherical Limit Points: linking three distinct areas"

Let the group G act by isometries on a proper CAT(0) space M, and let A be a finitely generated ZG-module. With this double action of G (geometric on M and algebraic on A) comes the set of horospherical limit points of the module A over the space M. It is a subset of the visual boundary of M.

The horospherical limit set has interesting interpretations in several areas:

    (1) in the case when M is Gromov-hyperbolic it gives a geometric criterion for deciding when A is finitely generated over a given normal subgroup of G;
    (2) in the flat case, where M is a Euclidean space and G is finitely generated free abelian acting by translations, it is (the integer analog of) the tropicalization of a certain algebraic variety;
    (3) in the case of G = SL(n,Z) acting on its symmetric space, it appears, at least conjecturally, as an interesting associated building when the boundary is retopologized by the Tits metric.

I will talk about these ideas.

This is joint work with Robert Bieri, growing out of his original work on the Bieri-Neumann-Strebel invariant (which I’ll define).