Smith theory, L2 cohomology, isometries of locally symmetric manifolds and moduli spaces of curves

TL;DR

This paper explores the symmetry properties of non-compact aspherical manifolds, especially moduli spaces of curves and locally symmetric spaces, showing they have limited homotopically trivial periodic diffeomorphisms and maximal symmetry under their natural metrics.

Contribution

It establishes new results on the absence of trivial periodic diffeomorphisms in certain spaces and proves an analogue of Royden's theorem for moduli spaces with finite volume.

Findings

Certain spaces have no homotopically trivial periodic diffeomorphisms.

Complete metrics on irreducible locally symmetric spaces have maximal symmetry.

An analogue of Royden's theorem is proven for moduli spaces of curves.

Abstract

We investigate periodic diffeomorphisms of non-compact aspherical manifolds (and orbifolds) and describe a class of spaces that have no homotopically trivial periodic diffeomorphisms. Prominent examples are moduli spaces of curves and aspherical locally symmetric spaces with non-vanishing Euler characteristic. In the irreducible locally symmetric case, we show that no complete metric has more symmetry than the locally symmetric metric. In the moduli space case, we build on work of Farb and Weinberger and prove an analogue of Royden's theorem for complete finite volume metrics.