Document Type

Dissertation

Date of Degree

Summer 2017

Access Restrictions

Access restricted until 08/31/2019

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Ionut Chifan

Abstract

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.

Keywords

Hyperbolic group, Prime, Rigidity, von Neumann Algebra

Pages

vii, 111 pages

Bibliography

Includes bibliographical references (pages 106-111).

Copyright

Copyright © 2017 Rolando de Santiago

Available for download on Saturday, August 31, 2019

Included in

Mathematics Commons

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