This work is a compilation of structural results for the von Neumann algebras of poly-hyperbolic groups established in a series of works done jointly with I. Chifan and T. Sinclair; and S. Pant. These works provide a wide range of circumstances where the product structure, a discrete structural property, can be recovered from the von Neumann algebra (a continuous object).
The primary result of Chifan, Sinclair and myself is as follows: if Γ = Γ1 × · · · × Γn is a product of non-elementary hyperbolic icc groups and Λ is a group such that L(Γ)=L(Λ), then Λ decomposes as an n-fold product of infinite groups. This provides a group-level strengthening of the unique prime decomposition of Ozawa and Popa by eliminating any assumption on the target group Λ. The methods necessary to establish this result provide a malleable procedure which allows one to rebuild the product of a group from the algebra itself.
Modifying the techniques found in the previous work, Pant and I are able to demonstrate that the class of poly-groups exhibit a similar phenomenon. Specifically, if Γ is a poly-hyperbolic group whose corresponding algebra is non-prime, then the group must necessarily decompose as a product of infinite groups.
| Identifer | oai:union.ndltd.org:uiowa.edu/oai:ir.uiowa.edu:etd-7212 |
| Date | 01 August 2017 |
| Creators | de Santiago, Rolando |
| Contributors | Chifan, Ionut |
| Publisher | University of Iowa |
| Source Sets | University of Iowa |
| Language | English |
| Detected Language | English |
| Type | dissertation |
| Format | application/pdf |
| Source | Theses and Dissertations |
| Rights | Copyright © 2017 Rolando de Santiago |
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