Convergence of spherical averages for Fuchsian groups
Vortrag von Prof. Dr. Caroline Series
Datum: 14.05.25 Zeit: 13.30 - 14.30 Raum: ETH HG F 5
Given a measure preserving action of a group G on a probability space X and a real valued function f on X, we consider the spherical averages S_n(f) of the functions f(g.x) averaged over all elements g of length n in a fixed set of generators. The limiting behaviour of S_n(f) has long been studied. Cesaro convergence has been proved in a wide variety of contexts. Actual convergence (depending on the parity of n) for free groups was proved by Nevo-Stein for f in L^p, p>1 In 2002, Bufetov extended the Nevo-Stein result to a slightly wider class by using a certain self-adjointness property of an associated Markov operator, which in turn depends on the fact that the inverse of a reduced word in a free group is itself reduced. In this talk we explain the same result for a large class of Fuchsian groups with presentations whose relations all have even length. The method relies on a new twist on the Bowen-Series coding for Fuchsian groups: by encoding the set of all shortest words representing a particular group element simultaneously, we obtain a suitable self-adjointness property of an associated Markov operator to which we apply a variant of Bufetov's original proof. This is joint work with Alexander Bufetov and Alexey Klimenko. [Published in Comm Math Helv 2023]