Convergence of spherical averages for measure-preserving actions of Fuchsian groups

The talk is based on the joint result with C. Series (Warwick University, UK). We show the pointwise convergence of spherical averages for measure-preserving actions of Fuchsian groups in a wide class (namely, those satisfying the so-called "even corners condition"). This result is a generalization of the similar result for the actions of free groups obtained by A. Bufetov in 2001. The basis for the proof is the construction of a new Markov coding for the Fuchsian group that satisfies the following symmetry condition: consider a Markov path corresponding to a group element $g$, apply a certain involution to each state in this path and write them in backward order; then this sequence is also a path in our Markov chain and it corresponds to $g^{-1}$. Our coding is the first symmetric coding for a wide class of group actions.
The construction is based on a consideration of the thickened paths, that is, the union of all shortest paths between the given pair vertices in the Cayley graph. The thickened path is split into levels, which are the collection of vertices on a given distance from its beginning. We show that if the geometric configuration of two adjacent levels is endowed by some additional data, these configurations form a topological Markov chain.

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