### Critical exponents for normal subgroups via a twisted Bowen-Margulis current

Ergodic Theory and Dynamical Systems Seminar

4th October 2018, 2:00 pm – 3:00 pm

Howard House, 4th Floor Seminar Room

For a discrete group $\Gamma$ of isometries of a negatively curved space $X$, the critical exponent $\delta(\Gamma)$ measures the exponential growth rate of the orbit of a point. It is known for a certain class of $\Gamma_0$ and $X$, that for any normal subgroup $\Gamma$ of $\Gamma_0$, we have $\delta(\Gamma)=\delta(\Gamma_0)$ if and only if the quotient $\Gamma_0/\Gamma$ is amenable. We will motivate this problem, and discuss what is new: the construction of a twisted Bowen-Margulis current on the double-boundary, which highlights a feature of ergodicity, and extends the class for which the result is known. This is joint work with R. Coulon, B. Schapira and S. Tapie.

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