Characterizing hyperelliptic surfaces in terms of closed geodesics

Abstract

We give a characterization of hyperelliptic surfaces in terms of simple, closed geodesics on the surfaces and graphs associated to these geodesics.

A geometric characterization of cyclic p-gonal surfaces

“…In this paper, cyclic p-gonal surfaces are characterized in terms of trees corresponding to p-ovals (or p-trees), which are collections of simple geodesic arcs contained in S (see Section 4). Although the present work mirrors the results of [3] and is intended as a companion paper, there is one basic difference: involutions on surfaces of genus g ≥ 2 can be determined by their action on certain subsurfaces. This is not the case with arbitrary automorphisms of order p > 2 (compare [3], Theorem 2 and its proof, with Proposition 3.4 in this paper).…”

“…The category includes hyperelliptic surfaces, where p = 2. The purpose of this paper is to give a geometric characterization of cyclic p-gonal surfaces for p > 2 akin to that given by the author in [3] for hyperelliptic surfaces.…”

“…In [3], hyperelliptic surfaces are characterized in terms of trees corresponding to simple, closed geodesics. In this paper, cyclic p-gonal surfaces are characterized in terms of trees corresponding to p-ovals (or p-trees), which are collections of simple geodesic arcs contained in S (see Section 4).…”

A geometric characterization of cyclic p-gonal surfaces

“…The next result follows from Proposition 3.1 and Lemma 2.1 in [3]. Proposition 3.2 Let S = U/G be a compact surface of genus g ≥ 2, with G a Fuchsian group, and let H be a non-elementary, precisely embedded subgroup of G of type (g − n, q), 0 < n ≤ g, 0 < q.…”

“…, l d , d > 0, contained in S, such that ∪ d 1 l i is the boundary in S of R. The next proposition follows easily from the definitions and the fact that p H is a covering map. As in [3], we omit the proof.…”