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

**Time:** 2018.01.16, 15:15–17:00

**Location:** Seminarraum 9, Oskar-Morgenstern-Platz 1, 2.Stock

**Title:** "Critical exponents for normal subgroups."

**Speaker:** Rhiannon Dougall (Université de
Nantes)

**Abstract:**
Fix a cocompact group \(\Gamma_0\) of isometries of a negatively curved, simply connected manifold \(X\). We are interested in the dynamics of its normal subgroups \(\Gamma\). Namely, we study the critical exponent \(\delta_\Gamma\), which is the exponential growth rate of the \(\Gamma\)–orbit of a point. We characterise the existence of a gap \(\delta_\Gamma<\delta_{\Gamma_0}\) uniform in a family of normal subgroups \(\Gamma\), in terms of permutation representations given by the quotients \(\Gamma_0/\Gamma\).
The proof uses the symbolic dynamics for the geodesic flow, for which we obtain the analogous statements for countable state shifts obtained as group extensions of a finite state shift.