# Hyperbolic groups with boundary an n-dimensional Sierpinski space

with J. Lafont

### J. Topol. Anal. 11 (2019) 1, 233–247.

The fundamental group of a closed aspherical manifold M is an example of a Poincare duality group. Whether or not all finitely presented Poincare duality groups arise in this fashion is an open problem that goes back to Wall. This paper addresses a relative version of this problem for a special class of groups. Fixing n>6, we show that if G is a torsion-free hyperbolic group whose visual boundary is an (n-2)-dimensional Sierpinski space, then G = pi_1(W) for some aspherical n-manifold W with boundary. The proof involves the total surgery obstruction and is based on work of Bartels-Lueck-Weinberger. Concerning the converse, we construct examples of aspherical manifolds with boundary whose fundamental group is hyperbolic but with visual boundary not homeomorphic to a Sierpinski space.