Pure Maths Colloquium: Richard Sharp

It is well-known that the number of periodic orbits of a geodesic flow over a compact negatively curved manifold (or, more generally, of an Anosov flow) grows exponentially fast. We are interested in the growth rate. If we pass to a finite regular cover then the growth rate remains the same but if the cover is infinite then, with an appropriate definition, the growth rate may decrease. It then becomes interesting to characterise when we still have equality. For geodesic flows, a combination of work of Roblin (2005) and Dougall and myself (2016) establishes that equality holds if and only if the covering group is amenable. A key feature underpinning this result if the time-reversal symmetry enjoyed by geodesic flows and the result fails for general Anosov flows, when this symmetry is absent. We will discuss these topics and a recent result that provides a natural generalisation to the Anosov setting. This is joint work with Rhiannon Dougall.