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

First Committee Member

Richard Baker

Second Committee Member

Victor Camillo

Third Committee Member

Paul Muhly

Fourth Committee Member

Gerhard Strohmer

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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