The Linear Part of a Discontinuously Acting Euclidean Semigroup

✍ Scribed by G.A. Soifer

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 146 KB
• Volume: 250
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.9054

✦ Synopsis

Let n be a Euclidean space and let S be a Euclidean semigroup, i.e., a subsemigroup of the group of isometries of n . We say that a semigroup S acts discontinuously on n if the subset s ∈ S sK ∩ K = is finite for any compact set K of n . The main results of this work are Theorem. If S is a Euclidean semigroup which acts discontinuously on n , then the connected component of the closure of the linear part S of S is a reducible group.

Corollary. Let S be a Euclidean semigroup acting discontinuously on n ; then the linear part S of S is not dense in the orthogonal group O n .

These results are the first step in the proof of the following Margulis' Conjecture. If S is a crystallographic Euclidean semigroup, then S is a group.