Orientation-reversing free actions on closed orientable surfaces by finite groups

Craig Merle Johnson, Purdue University

Abstract

Free actions by finite groups on surfaces U$\sp{\rm k}$ = RP$\sp2 \# \...\#$ RP$\sp2$ (k times) and T$\sp{\rm k}$ = T # T # ... # T (k times), where T = S$\sp1$ $\times$ S$\sp1$, were studied by L. Cusick in (1). For $\vert$G$\vert$ = 2n, let $\partial$: $\pi\sb1$U$\sp{\rm r+2}$ $\to$ G be an epimorphism. Define N$\sb\partial$ to be the image under $\partial$ of the even words in $\pi\sb1$U$\sp{\rm r+2}$ (where "even" is well-defined under the standard presentation for $\pi\sb1$U$\sp{\rm r+2}$). In (1) it is shown that if (G: N$\sb\partial$) = 2, then ker $\partial$ is isomorphic to $\pi\sb1$T$\sp{\rm nr+1}$. In this paper the converse is proven to be true, answering a conjecture by Cusick, and the resulting theorem provides a convenient tool for continuing the study of free actions on the closed orientable surfaces T$\sp{\rm k}$. Namely, (G: N$\sb\partial$) = 2 iff ker$\partial$ $\cong\pi\sb1$T$\sp{\rm nr+1}$ iff there exists an exact sequence$$\rm 1\to\pi\sb1 T\sp{nr+1} \to \pi\sb1 U\sp{r+2}{\buildrel\partial\over\longrightarrow} G \to 1.$$This sequence, in turn, is associated with an orientation-reversing free action by G on T$\sp{\rm nr+1}$. The main results are: (1) The determination of upper bounds for the numbers of orientation-reversing free actions on T$\sp{\rm k}$ by finite groups of order less than or equal to 14. (2) The establishment of values for n and r determining when the finite cyclic group Z$\sb{\rm 2n}$ act freely, orientation-reversing, on T$\sp{\rm{nr+1}}$. (3) The development of algorithms which were used in computer programs to compute subgroups and determine epimorphisms.

Degree

Ph.D.

Advisors

Gottlieb, Purdue University.

Subject Area

Mathematics

COinS