## 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

#### 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

**Source***Probabilistic Approach to Geometry*, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 141-153

**Dates**

Received: 10 February 2009

First available in Project Euclid:
24 November 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/05710141

**Mathematical Reviews number (MathSciNet)**

MR2648257

**Zentralblatt MATH identifier**

1203.53033

**Subjects**

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]

**Keywords**

Dirichlet sum of order $p$ Royden compactification quasi isometry

#### 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