Mathematical Proceedings of the Cambridge Philosophical Society


Research Article

Upper and lower bounds for the mass of the geodesic flow on graphs


MICHEL COORNAERT a1 and ATHANASE PAPADOPOULOS a1
a1 Institut de Recherche Mathématique Avancée (Université Louis Pasteur et CNRS), 7 rue René Descartes, 67084 Strasbourg Cedex (France)

Abstract

Let G be a connected locally finite simplicial graph with rk([pi]1(G))[gt-or-equal, slanted]2 and let T be the universal cover of G. Consider a [pi]1(G)-invariant conformal density [mu] of dimension d on [partial partial differential]T. The total mass function [phi][mu] of [mu] is defined on the set of vertices of G. Let |[phi][mu]| be its l2-norm. Let [Omega] be the geodesic flow space of G and m[mu] the invariant measure on [Omega] associated to [mu]. The main results of this paper are the following:

(i) Assume that there exists an integer k[gt-or-equal, slanted]3 such that the degree at each vertex of G is [less-than-or-eq, slant]k. Then, if d>[fraction one-half]log(k[minus sign]1) and m[mu]([Omega])<[infty infinity], we have [phi][mu][set membership]l2(V) and

|[phi][mu]|2 [less-than-or-eq, slant]Cm[mu]([Omega]),

with C= (e2d[minus sign]1)/ (e2d[minus sign]k+1).

(ii) Assume that there exists an integer k[gt-or-equal, slanted]3 such that the degree at each vertex of G is [gt-or-equal, slanted]k. Then, if [phi][mu][set membership]l2(V), we have d>[fraction one-half]log(k[minus sign]1) and

Cm[mu]([Omega]) [less-than-or-eq, slant]|[phi][mu]|2.

(Received October 11 1995)
(Revised January 23 1996)