On the automorphism groups of hyperbolic manifolds
Abstract
Let Diff(N) and Homeo(N) denote the smooth and topological group of automorphisms respectively that fix the boundary of the nmanifold N, pointwise. We show that the (n4)th homotopy group of Homeo(S^1 \times D^{n1}) is not finitelygenerated for n >= 4 and in particular the topological mappingclass group of S^1\times D^3 is infinitely generated. We apply this to show that the smooth and topological automorphism groups of finitevolume hyperbolic nmanifolds (when n >= 4) do not have the homotopytype of finite CWcomplexes, results previously known for n >= 11 by Farrell and Jones. In particular, we show that if N is a closed hyperbolic nmanifold, and if Diff_0(N) represents the subgroup of diffeomorphisms that are homotopic to the identity, then the (n4)th homotopy group of Diff_0(N) is infinitely generated and hence if n=4, then \pi_0\Diff_0(N) is infinitely generated with similar results holding topologically.
 Publication:

arXiv eprints
 Pub Date:
 March 2023
 DOI:
 10.48550/arXiv.2303.05010
 arXiv:
 arXiv:2303.05010
 Bibcode:
 2023arXiv230305010B
 Keywords:

 Mathematics  Geometric Topology;
 57M99;
 57R52;
 57R50;
 57N50
 EPrint:
 32 pages, 17 figures