Coarse geometry and P. A. Smith theory (Q719065)

The classical P. A. Smith theory is one of the most important tools in the cohomology theory of topological transformation groups. As an application of this theory, \textit{G. E. Bredon} [Introduction to compact transformation groups. Pure and Applied Mathematics 46. New York-London: Academic Press (1972; Zbl 0246.57017)] proved that, if \(G\) is a \(p\)-group (\(p\) prime) and \(X\) is a finitistic \(G\)-space which is a mod \(p\) cohomology \(n\)-sphere, then the fixed point set \(X^G\) is a mod \(p\) cohomology \(r\)-sphere for some \(-1 \leq r \leq n\). Another well-known application of P. A. Smith theory is that \(\mathbb{Z}_p \oplus \mathbb{Z}_p\) cannot act freely on a finitistic mod \(p\) cohomology sphere. The purpose of this comprehensive paper is to give a coarse geometry analogue of P. A. Smith theory for proper geodesic metric spaces with finite asymptotic dimension in the sense of \textit{M. Gromov} [Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. London Mathematical Society Lecture Note Series. 182. Cambridge University Press (1993; Zbl 0841.20039)]. The authors introduce a coarse generalization of the usual fixed point set, called the bounded fixed set and make use of the coarse homology as defined by \textit{J. Roe} [Coarse cohomology and index theory on complete Riemannian manifolds. Mem. Am. Math. Soc. 497 (1993; Zbl 0780.58043)]. The paper contains many interesting results and the proposed theory is suitable for applications in geometric group theory. The two main results are the following: Theorem A. Let \(X\) be a proper geodesic metric space with finite asymptotic dimension, which is a mod \(p\) coarse homology \(m\)-sphere, for some prime \(p\). Let \(G\) be a finite \(p\)-group with a tame action on \(X\) by isometries. Then the bounded fixed set \(X_{bd}^G\) is a mod \(p\) coarse homology \(r\)-sphere, for some \(0 \leq r \leq m\). If \(p\) is odd, then \(m-r\) is even. Theorem B. The group \(G=\mathbb{Z}_p \oplus \mathbb{Z}_p\), for \(p\) a prime, cannot act tamely and semifreely at the large scale on a mod \(p\) coarse homology \(m\)-sphere \(X\), whenever \(X\) is a proper geodesic metric space with finite asymptotic dimension, and \(X_{bd}^G\) is a mod \(p\) coarse homology \(r\)-sphere, for some \(0 \leq r < m\).