**SMS scnews item created by Stephan Tillmann at Mon 8 Sep 2014 1025**

Type: Seminar

Distribution: World

Expiry: 8 Dec 2014

**Calendar1: 10 Sep 2014 1100-1200**

**CalLoc1: Carslaw 535A**

Auth: tillmann@p710.pc (assumed)

### Geometry-Topology-Analysis Seminar

# A Multiplicative Ergodic Theorem for p-adic Lie groups

### James Parkinson

GTA Seminar - Wednesday, 10 September, 11:00-12:00 in Carlaw 535A

Please join us for lunch after the talk!

##
A Multiplicative Ergodic Theorem for p-adic Lie groups

James Parkinson (Sydney)

Abstract

The celebrated Multiplicative Ergodic Theorem of Oseledets shows
that under a finite first moment assumption, the product of random real iid
matrices behaves asymptotically like the sequence of powers of some fixed
positive definite symmetric matrix. In 1989 Vadim Kaimanovich showed that
this property can be expressed in purely geometric terms using the symmetric
space associated to \(GL_n(R)\). This observation lead to the notion of a
'regular sequence' in a symmetric space, and Kaimanovich gave a complete
characterisation of these sequences in terms of spherical and horospheric
coordinates in the symmetric space. As a consequence of this
characterisation Kaimanovich obtained a Multiplicative Ergodic Theorem for
noncompact semisimple real Lie groups with finite centre, generalising the
original theorem of Oseledets.

In this talk we will discuss a p-adic analogue of this story. In this
setting the symmetric space is replaced by the affine building of the p-adic
group. We define regular sequences in affine buildings, and give a
characterisation of these sequences in terms of analogues of the spherical
and horospherical coordinates from the real theory. We then discuss
applications to a Multiplicative Ergodic Theorem for Lie groups defined over
p-adic fields. This is joint with W. Woess.