Calendar

Please join us for pre-seminar Tea

Confined subgroups of a Lie group or of a discrete group are generalizations of a Fuchsian group G that gives rise to a hyperbolic surface M of bounded injectivity radius. When the surface M has finite volume, the growth rate of G equals the volume growth rate of the hyperbolic plane H^2. Gekhtman and Levit established an inequality between the growth rates of G and the ambient space for a general confined subgroup of a rank-1 Lie group. In this talk, I will explain an analogous result for a confined subgroup G of a hyperbolic group, rank-1 CAT(0) group, or the mapping class group. This is established by studying the single (as opposed to the double) boundary action of G. Joint work with Ilya Gekhtman, Wenyuan Yang, and Tianyi Zheng.

Continuous cohomology classes of the group GL_n(Z_p) with coefficients in Q_p give rise to a theory of characteristic classes for étale Z_p-local systems on algebraic varieties, valued in (absolute) étale cohomology with Q_p-coefficients. These classes can be thought of as p-adic analogs of Chern–Simons characteristic classes of complex vector bundles with a flat connection.

By a theorem of Reznikov, Chern–Simons classes of all complex local systems on a smooth proper algebraic variety are torsion in degrees > 1. The same turns out to be true for p-adic characteristic classes on smooth varieties over algebraically closed fields (at least for p large enough, as compared to the rank of the local system). But for varieties over number fields and local fields these p-adic characteristic classes happen to be non-trivial even rationally, and for local systems coming from cohomology of a family of algebraic varieties they can be partially expressed in terms of the Chern classes of the corresponding Hodge bundles. 

This computation of p-adic characteristic classes relies on the notion of a Chern class for pro-étale vector bundles, and on the Hodge–Tate filtration in relative p-adic Hodge theory. This is joint work with Lue Pan.