# Symmetries of exotic negatively curved manifolds

## TL;DR

This paper investigates the symmetry properties of manifolds that are topologically but not smoothly equivalent to hyperbolic manifolds, providing examples with both maximal and minimal symmetry groups.

## Contribution

It constructs examples of exotic negatively curved manifolds exhibiting either maximal or minimal symmetry, using advanced techniques like smoothing theory and hyperbolic manifold construction.

## Key findings

- Existence of manifolds with arbitrarily large symmetry groups.
- Existence of manifolds with minimal symmetry, no small subgroup acts by diffeomorphisms.
- Demonstrates the range of symmetry behaviors in exotic negatively curved manifolds.

## Abstract

Let $N$ be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold $M$. In this paper, we study the extent to which $N$ admits as much symmetry as $M$. Our main results are examples of $N$ that exhibit two extremes of behavior. On the one hand, we find $N$ with maximal symmetry, i.e. Isom($M$) acts on $N$ by isometries with respect to some negatively curved metric on $N$. For these examples, Isom($M$) can be made arbitrarily large. On the other hand, we find $N$ with little symmetry, i.e. no subgroup of Isom($M$) of "small" index acts by diffeomorphisms of $N$. The construction of these examples incorporates a variety of techniques including smoothing theory and the Belolipetsky-Lubotzky method for constructing hyperbolic manifolds with a prescribed isometry group.