Hyperbolic unit groups

Abstract

In this paper we study the groups G {\mathcal {G}} whose integral group rings have hyperbolic unit groups U ( Z G ) {\mathcal {U}(\mathbb {Z} {\mathcal {G}}) } . We classify completely the torsion subgroups of U ( Z G ) \mathcal {U}(\mathbb {Z} {\mathcal {G}}) and the polycyclic-by-finite subgroups of the group G {\mathcal {G}} . Finally, we classify the groups for which the boundary of U ( Z G ) {\mathcal {U}(\mathbb {Z} {\mathcal {G}}) } has dimension zero.