Event time:

Thursday, October 5, 2023 - 4:00pm

Location:

KT801

Speaker:

Pierre-Louis Blayac

Speaker affiliation:

University of Michigan

Event description:

In the theory of discrete subgroups of Lie groups, given a length function on the Lie group G, one popular object of study is the asymptotic when R goes to infinity of the number of loxodromic elements with length less R in a discrete subgroup H, and the distribution of the fixed points of these loxodromic elements.

When G is the group of (real) projective transformations and H acts properly discontinuously and cocompactly on a strictly convex domain O of the projective space, Yves Benoist noticed that one can use the Hilbert metric of O, and the associated Hilbert geodesic flow, to estimate the above counting function, for a special length function called the Hilbert length.

An important ingredient in Benoist’s result is the uniform hyperbolicity of the Hilbert geodesic flow, which does not hold when O is not strictly convex.

In this talk we will develop and use instead the theory of Patterson–Sullivan measures for Hilbert geometries O/H that satisfy a mild rank-one assumption, and are not necessarily strictly convex nor compact.

This will yield counting results for H in the cases where the induced Bowen–Margulis measure is finite. (This is joint work with Feng Zhu).