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Analysis on Riemannian co-compact covers. (English) Zbl 1082.31006

Grigor’yan, Alexander (ed.) et al., Surveys in differential geometry. Eigenvalues of Laplacians and other geometric operators. Somerville, MA: International Press (ISBN 1-57146-115-9/hbk). Surveys in Differential Geometry 9, 351-384 (2004).
We have here a survey about potential analytic properties of a special class of Riemannian manifolds: the non compact co-compact Riemannian covers. These manifolds are closely related with Cayley graphs and this relation inspires in a great extent the work.
If, given a Riemannian manifold \(M\), there exists a discrete subgroup \(\Gamma\) of the group of isometries of \(M\) such that \(N=M/\Gamma\) is a compact Riemannian manifold, then we say that \(M\) is a regular (co-compact) cover of \(N\) with deck transformation group \(\Gamma\).
There are two fundamental observations:
(1) A Riemannian cover \(M\) of a compact manifold with deck transformation group \(\Gamma\) is quasi-isometric, (as a metric space), to any fixed Cayley graph \((\Gamma, S)\), (\(S\) being the generating set).
(2) The local geometry of \(M\) is uniformly under control, in the sense that the volume function satisfies the doubling property as long as the radius of the balls are bounded above, the balls of radius \(1\) all have comparable volume, and uniform local Harnack inequalities (elliptic and parabolic) are satisfied.
Another category of spaces with its local geometry uniformly under control are the bounded geometry (BG) spaces. A BG space is either a complete Riemannian manifold with Ricci curvature bounded below and with a uniform lower bound on the volume balls of radius \(1\) or a finite connected graph (equipped with its graph distance and the measure given by the degree of the vertex) with uniformly bounded degree. It can be proved that any manifold in BG is quasi isometric to some graph in BG.
The survey offers a revision of some results concerning geometric and potential theoretic properties of all such kind of spaces, namely, properties related with heat kernel estimates, parabolicity, volume growth, isoperimetry, Dirichlet spectrum and Liouville properties.
The key idea along all the work is that these geometric and potential theoretic properties, considered on co-compact regular covers, are determined by the behaviour of some random walks on its deck transformation group.
For the entire collection see [Zbl 1050.53002].

MSC:

31C12 Potential theory on Riemannian manifolds and other spaces
58J35 Heat and other parabolic equation methods for PDEs on manifolds
20F69 Asymptotic properties of groups
60G50 Sums of independent random variables; random walks
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