CLOSED GEODESICS ON COMPACT DEVELOPABLE ORBIFOLDS

Abstract

<p>Existence of closed geodesics on compact manifolds was first proved by Lyusternik and Fet in the 1950s using Morse theory, and the corresponding problem for orbifolds was studied by Guruprasad and Haefliger, who proved existence of a closed geodesic of positive length in numerous cases. In this thesis, we develop an alternative approach to the problem of existence of closed geodesics on compact orbifolds by studying the geometry of group actions. We give an independent and elementary proof that recovers and extends the results of Guruprasad and Haefliger for developable orbifolds. We show that every compact orbifold of dimension 2, 3, 5 or 7 admits a closed geodesic of positive length, and we give an inductive argument that reduces the existence problem to the case of a compact developable orbifold of even dimension whose singular locus is zero-dimensional and whose orbifold fundamental group is infinite torsion and of odd exponent. Stronger results are obtained under curvature assumptions. For instance, one can show that infinite torsion groups do not act geometrically on simply connected manifolds of nonpositive or nonnegative curvature, and we apply this to prove existence of closed geodesics for compact orbifolds of nonpositive or nonnegative curvature. In the general case, the problem of existence of closed geodesics on compact orbifolds is seen to be intimately related to the group-theoretic question of finite presentability of infinite torsion groups, and we explore these and other properties of the orbifold fundamental group in the last chapter.</p>