Contracting automorphisms and L^p-cohomology in degree one

We characterize those Lie groups, and algebraic groups over a local field of characteristic zero, whose first reduced L^p-cohomology is zero for all p>1, extending a result of Pansu. As an application, we obtain a description of Gromov-hyperbolic groups among those groups. In particular we prove that any non-elementary Gromov-hyperbolic algebraic group over a non-Archimedean local field of zero characteristic is quasi-isometric to a 3-regular tree. We also extend the study to semidirect products of a general locally compact group by a cyclic group acting by contracting automorphisms.Comment: 27 pages, no figur

Maximal Compact Subgroups In Locally Compact Groups (lie)

The main concern is the existence of maximal compact normal subgroup K in a locally compact topological group G, and whether or not G/K is a Lie group. If G is a locally compact group, then G has maximal compact subgroup if and only if G/G(,o) has. Every compact subgroup of a totally disconnected group is contained in an open compact subgroup. As a result, maximal compact subgroups of totally disconnected groups are open. If the bounded part of G is compactly generated, then G has a maximal compact normal subgroup. Generalized FC-groups, compactly generated type I IN-groups, and Moore groups have this desired property. If P(G/G(,o)) is a compact subgroup of G, then G has a maximal compact normal subgroup with Lie factor group. Compact extensions of compactly generated nilpotent groups have maximal compact normal subgroups with Lie factor groups. Examples and counterexamples show limitations on extending these results.The next central problem is the approximation of locally compact groups by Lie groups using inverse limits. The inverse limit of Lie groups, if it is locally compact, is a pro-Lie group. Conversely, a pro-Lie group is an inverse limit of Lie groups. The inverse limit of locally compact groups may not be locally compact. A class more general than pro-Lie groups, namely, strongly residual Lie groups is discussed. A residual Lie group can be approximated by finite dimensional factor groups of compact connected normal subgroups