# L^2-Betti number through triangle counting

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Léo Bénard

created by collari on 18 Oct 2023

modified on 20 Nov 2023

27 nov 2023
-- 14:30

**Abstract.**

Given a compact manifold of dimension n with a triangulation, we define a combinatorial zeta function associated to random walks in the (n-1) skeleton of the triangulation. We show that this series extends as an analytic function on some disk, and recover a topological invariant, the first L^{2}-Betti number, as its vanishing order at some point in the boundary of this disk.