## Advanced Studies in Pure Mathematics

### Functions of finite Dirichlet sums and compactifications of infinite graphs

#### Abstract

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.

#### Article information

Dates
First available in Project Euclid: 24 November 2018

https://projecteuclid.org/ euclid.aspm/1543086316

Digital Object Identifier
doi:10.2969/aspm/05710141

Mathematical Reviews number (MathSciNet)
MR2648257

Zentralblatt MATH identifier
1203.53033

#### Citation

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