Generalized spherical space forms.

Abstract

In differential geometry, the pseudo-spherical space form is defined by $S\sp{n,m}$, where $\Gamma$ is a discrete subgroup of $O(n + 1, m)$ acting freely and properly discontinuously on the pseudosphere $S\sp{n,m}$ $\cong$ $S\sp n \times \IR\sp m$. One would like to determine and classify all such $\Gamma$. In Chapter I, we consider a finite subgroup $\Phi$ of $O(n + 1, m)$ and we show that if $\Phi$ acts freely on $S\sp n \times \IR\sp m$, then $\Phi$ is isomorphic to a subgroup of $O(n + 1)$ and hence it satisfies the pq-conditions. Also, we show that the fixed point set ($S\sp n \times \IR\sp m)\sp{\Phi}$ by a finite subgroup $\Phi \subseteq O(n + 1, m)$ is isometric to $S\sp{n\sp\prime, m\sp\prime}$ for some 0 $\leq n\sp\prime \leq n$ and 0 $\leq m\sp\prime \leq m$. In Chapter II, by considering the Seifert fiber space construction modeled on the product space $P = S\sp3 \times \IR\sp m$, we study discrete subgroups of Top$\sb{S\sp3}(P)$, the topological group of weakly $S\sp3$-equivariant homeomorphisms. We prove that if $\Gamma$ is an extension of a finite subgroup $\Phi$ of $S\sp3$ by a discrete group $Q$ acting properly discontinuously on $\IR\sp m$ and if $\Gamma$ acts on $S\sp3 \times \IR\sp m$ as a group of weakly $S\sp3$-equivariantly homeomorphisms, then $\Gamma$ must be a subgroup of $S\sp3 \times Q$ and vice versa. The geometric problem which is discussed in Chapter I motivates the study of the topological symmetries of the product space $S\sp n \times \IR\sp m$. A good deal is known about group actions on $S\sp n$ and $\IR\sp m$. We consider some questions concerning the groups that act freely and properly discontinuously on $S\sp n \times \IR\sp m$ and which generalize properties of groups acting freely and properly discontinuously on $S\sp n$ or $\IR\sp m$. In Chapter III, we prove that if a discrete group $\Gamma$ acts freely and properly discontinuously on $S\sp{n}\times\IR\sp{m}$ and if $vcd(\Gamma)$ $<$ $\infty$, then $vcd(\Gamma)$ $\leq m$ with equality if and only if $\Gamma\\(S\sp{n}\times\IR\sp{m})$ is compact. Moreover, if $\Gamma\\(S\sp{n}\times\IR\sp{m})$ is compact, then $\Gamma$ has a finite index subgroup $\Gamma\sb0$ which is a Poincare duality group of dimension $m$. If $n$ is even we show that $\Gamma$ is virtually torsion free; in fact, $\Gamma$ is torsion free, or else $\Gamma$ is isomorphic to $\Gamma\sb0\times\doubz\sb2$, where $\Gamma\sb0$ is torsion free. Also we prove that $\Gamma$ has periodic Farrell cohomology. Its period is 2 if $\Gamma$ has some torsion and $n$ is even. If $vcd(\Gamma)$ $<$ $\infty$ and $n$ is odd, then the period of $\Gamma$ is at most $n+1$.