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Distributional limits of Riemannian manifolds and graphs with sublinear genus growth. (English) Zbl 1291.53056

Summary: I. Benjamini and O. Schramm [Electron. J. Probab. 6, Paper No. 23, 13 p. (2001; Zbl 1010.82021)] introduced the notion of distributional limit of a sequence of graphs with uniformly bounded valence and studied such limits in the case that the involved graphs are planar. We investigate distributional limits of sequences of Riemannian manifolds with bounded curvature which satisfy a quasi-conformal condition. We then apply our results to somewhat improve Benjamini’s and Schramm’s original result on the recurrence of the simple random walk on limits of planar graphs. For instance, as an application we give a proof of the fact that for graphs in an expander family, the genus of each graph is bounded from below by a linear function of the number of vertices.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C20 Global Riemannian geometry, including pinching
05C10 Planar graphs; geometric and topological aspects of graph theory

Citations:

Zbl 1010.82021
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References:

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