Advanced Studies in Pure Mathematics
- Adv. Stud. Pure Math.
- Probabilistic Approach to Geometry, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 141 - 153
Functions of finite Dirichlet sums and compactifications of infinite graphs
We introduce the $p$-resister for an infinite network and show a comparison theorem on the resisters for two infinite graphs of bounded degrees which are quasi isometric. Some geometric projections of the Royden $p$-compactifications of infinite networks are investigated and several observations are made in relation to geometric boundaries of hyperbolic networks in the sense of Gromov. In addition, a Riemannian manifold which is quasi isometric to the hyperbolic space form is constructed to illustrate a role of the bounded local geometry in studying points at infinity.
Received: 10 February 2009
First available in Project Euclid: 24 November 2018
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 58D17: Manifolds of metrics (esp. Riemannian) 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Hattori, Tae; Kasue, Atsushi. Functions of finite Dirichlet sums and compactifications of infinite graphs. Probabilistic Approach to Geometry, 141--153, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05710141. https://projecteuclid.org/euclid.aspm/1543086316