Symmetries of 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. $\operatorname{Isom}(M)$ acts on $N$ by isometries with respect to some negatively curved metric on $N$. For these examples, the order of $\operatorname{Isom}(M)$ can be arbitrarily large. On the other hand, we find $N$ with little symmetry, i.e. no subgroup of $\operatorname{Isom}(M)$ 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.