** "Geometry and Analysis on Groups" Research Seminar **

**Time:** 2019.01.29, 15:00–17:00

**Location:** Seminarraum 8, Oskar-Morgenstern-Platz 1, 2.Stock

**Title:** "Measure rigidity for horospherical subgroups of groups acting on regular trees."

**Speaker:** Vladimir Finkelshtein (Universität Göttingen)

**Abstract:**
Consider a locally compact group \(G\), and let \(H\lt G\) be a subgroup
acting on \(X=G/L\), where \(L\lt G\) is a lattice.
One of the
fundamental questions in homogeneous dynamics is to classify
\(H\)–invariant measures on \(X\). Often, one can then use the
knowledge about invariant measures in order to classify closures of
\(H\)–orbits in \(X\).
Such results are known under the general name of Ratner theorems and
have numerous applications in number theory and other fields.
Ratner theorems are abundant when \(G\) is a linear group.
Our focus is on groups acting on regular trees, which are generically
non-linear.
We prove that if \(G\) is a subgroup of the group of automorphisms of
a regular tree satisfying some technical conditions, \(L\lt G\) a
lattice, \(H\lt G\) a horospherical subgroup, then all
\(H\)–invariant measures on \(G/L\) are homogeneous.
Moreover, if \(L\) is a uniform lattice, we show unique ergodicity.
This is work in progress with Corina Ciobotaru and Cagri Sert.