Orientation-reversing free actions on handlebodies

Abstract

The observation that the quotient orbifold of an orientation-reversing involution on a 3-dimensional handlebody has the structure of a compression body leads to a strong classification theorem, and general structure theorems. The structure theorems decompose the action along invariant discs into actions on handlebodies which preserve the I-fibers of some I-bundle structure. As applications, various results of R. Nelson are proved without restrictive hypotheses. 0. Introduction Throughout this paper h will be an orientation-reversing involution on a 3-dimensional orientable handlebody H. Moreover, h is assumed to be tame, so the action of h on an invariant ball about a fixed point must be equivalent either to the antipodal map on the 3-ball, or to a reflection across a 2-disc in the 3-ball, or (precisely when the fixed point lies in ∂H) to a reflection across a half 2-disc in a half 3-ball. The fixed point set fix(h) is thus a union of finitely many isolated fixed points and finitely many 2-manifolds properly imbedded in H. It is known (The-orem 6.1 of [2]) that the 2-dimensional components are incompressible, although this will be seen in passing when we examine the structure of h, and is immediate from the structure theory we will develop. A simple type of involution occurs when h is a product h 1 × h 2 on A × I, where A is an orientable 2-manifold with boundary, each h i is either the identity or an involution, and exactly one of the h i is orientation-reversing. Also, when H is the orientable I-bundle over a nonorientable 2-manifold with boundary, one has similar examples. By gluing such actions together along invariant discs in the boundary, one constructs more complicated examples. Our main structure results, Theorems 5.1 and 5.6, show that all orientation-reversing involutions are built up in this way, and furnish descriptions of the component actions in terms of the fixed-point data of h and the genus of H. The actions on the pieces in these decompositions are sometimes but not always unique up to equivalence; the conditions under which uniqueness holds and the extent to which it fails are fully analyzed in auxiliary theorems. As corollaries of the main structure results, one obtains the following simple characterizations of fiber-preserving involutions in terms of the fixed point data.

A structural description of extended ${\mathbb Z}_{2n}$-Schottky groups

“…In [15] an explicit formulae to obtain the number of equivalence classes for cyclic groups of prime order p was provided. In [6] it was studied the case of free fixed point orientation-reversing group actions on handlebodies and a classification theorem was obtained in terms of algebraic invariants that involve Nielsen equivalence.…”