# Symmetries of exotic negatively curved manifolds

with M. Bustamante

### J. Diff. Geom. 120 (2022) 2, 231-250.

Let N be a smooth manifold that’s 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. 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.