Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8896605 | Journal of Functional Analysis | 2018 | 32 Pages |
Abstract
If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here “almost” and “close” are in terms of the Hilbert-Schmidt norm. Using tracial 2-norm on II1-factors we similarly define II1-factor stability for groups. Our main result is that all 1-relator groups with non-trivial center are II1-factor stable. Many of them are also matricially stable and RFD. For amenable groups we give a complete characterization of matricial stability in terms of the following approximation property for characters: each character must be a pointwise limit of traces of finite-dimensional representations. This allows us to prove matricial stability for the discrete Heisenberg group H3 and for all virtually abelian groups. For non-amenable groups the same approximation property is a necessary condition for being matricially stable. We study this approximation property and show that RF groups with character rigidity have it.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Don Hadwin, Tatiana Shulman,