Volume entropy of hyperbolic graph surfaces

Abstract

A graph surface P is a two-dimensional polyhedron having the simplest kind of non-trivial singularities which result from gluing surfaces with compact boundaries along boundary components. We study the behavior of the volume entropy h ( g ) of hyperbolic metrics g on a closed graph surface P depending on the lengths of singular geodesics $Q\subset P$ . We show that always h ( g ) > 1 and $h(g)\to\infty$ as $L_g(Q)\to\infty$ for at least one singular geodesic Q .