Infinite dimensional geometric groups

Author

Cornelius Stallmann

Date of Award

12-1996

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Conrad P. Plaut

Committee Members

Robert Daverman, Alexandre Freire, John Nolt

Abstract

Connected locally compact groups are generalized Lie groups. This allows us to bring the vast knowledge of Lie groups to bear on the study of such groups. For example the theory of covering groups generalizes to arcwise connected locally compact groups. However the results are perhaps somewhat unexpected and certainly not as satisfactory. The covering is no longer necessarily locally injective and the covering space is in general no longer locally compact. The kernel of the covering map is semidiscrete, (an inverse limit of discrete homogeneous spaces). Since quotients of arcwise connected groups by semidiscrete subgroups have strong lifting properties we would like to see under what conditions they occur as closed subgroups. We begin a characterization of arcwise connected groups that have the property that totally disconnected closed subgroups are always semidiscrete. We also show that semidiscrete groups are precisely those groups carrying a geometry over the semigroup R^max [6] ). In [6] it is shown that an invariant inner metric can be constructed for arcwise connected locally compact groups by restricting to a suitable "metric core" of the universal cover. This metric has useful properties which we investigate. We show that the metric can be adjusted so that a given one parameter subgroup is rectifiable. We show that each homotopy class of loops has a representative of minimal length. We show that the covering map when restricted to the metric core is a submetry, (a generalization of Riemannian submersion). We also show that it satisfies another property shared by traditional covering maps of inner metric spaces, the covering is a weak local isometry in that it preserves the lengths of curves. We investigate some interesting properties of the product metric on the infinite dimensional torus, T^∞. In contrast to a bi-invariant Riemannian metric on a compact Lie group this metric has one-parameter subgroups that are not geodesies as well as homotopy class without rectifiable representatives. This behavior is common to metrics in infinite dimensional groups. We give some results on the relationship between closed subgroups of the fundamental group of a group with universal cover, and the closed subgroups of the kernel. We show that an LP group with inner metric must be locally compact, confirming the need to restrict to the metric core in the universal cover.