Nonstandard hulls of locally uniform groups

Isaac Goldbring¹

¹ Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Science and Engineering Offices (M/C 249) 851 S. Morgan St. Chicago, IL 60607-7045, U.S.A.

Fundamenta Mathematicae, Tome 220 (2013) no. 2, pp. 93-118

Résumé

We present a nonstandard hull construction for locally uniform groups in a spirit similar to Luxemburg's construction of the nonstandard hull of a uniform space. Our nonstandard hull is a local group rather than a global group. We investigate how this construction varies as one changes the family of pseudometrics used to construct the hull. We use the nonstandard hull construction to give a nonstandard characterization of Enflo's notion of groups that are uniformly free from small subgroups. We prove that our nonstandard hull is locally isomorphic to Pestov's nonstandard hull for Banach–Lie groups. We also give some examples of infinite-dimensional Lie groups that are locally uniform.

Détail

DOI : 10.4064/fm220-2-1

Keywords: present nonstandard hull construction locally uniform groups spirit similar luxemburgs construction nonstandard hull uniform space nonstandard hull local group rather global group investigate construction varies changes family pseudometrics construct hull nonstandard hull construction nonstandard characterization enflos notion groups uniformly small subgroups prove nonstandard hull locally isomorphic pestovs nonstandard hull banach lie groups examples infinite dimensional lie groups locally uniform