DOI

10.17077/etd.dysxly3f

Document Type

Dissertation

Date of Degree

Summer 2017

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Chifan, Ionut

First Committee Member

Baker, Richard

Second Committee Member

Camillo, Victor

Third Committee Member

Jorgensen, Palle

Fourth Committee Member

Khurana, Surjit

Abstract

Chifan, Kida, and myself introduced a new class of non-amenable groups denoted by ${\bf NC} \cap {\bf Quot}(\mathcal C_{rss})$ which gives rise to \emph{prime} von Neumann algebras. This means that for every $\G\in {\bf NC} \cap {\bf Quot}(\mathcal C_{rss})$ its group von Neumann algebra $L(\G)$ cannot be decomposed as a tensor product of diffuse von Neumann algebras. The class ${\bf NC} \cap {\bf Quot}(\mathcal C_{rss})$ is fairly large as it contains many natural examples of groups, some intensively studied in various areas of mathematics: all infinite central quotients of pure surface braid groups; all mapping class groups of (punctured) surfaces of genus $0,1,2$; most Torelli groups and Johnson kernels of (punctured) surfaces of genus $0,1,2$; and, all groups hyperbolic relative to finite families of residually finite, exact, infinite, proper subgroups.

In a separate investigation, de Santiago and myself were able to extend the previous techniques that allowed us to eliminate the usage of the {\bf NC} condition and ultimately classify all the possible tensor factorization of the von Neumann algebras of groups that belong solely to ${\bf Quot}(\mathcal C_{rss})$. This provides a far-reaching generalization of the aforementioned primeness results; for instance, we were able to show that if $\Gamma$ is a poly-hyperbolic group, then whenever we have a tensor decomposition $L(\G)\cong P_1\bar\otimes P_2 \bar \otimes \cdots \bar\otimes P_n$ then there exists a product decomposition $\G\cong \G_1\times \G_2 \times \cdots \times \G_n$ with $\G_i \in {\bf Quot}(\mathcal C_{rss})$ and, up to amplifications, we have $L(\G_i)\cong P_i$ for all $i=1,n$.

Keywords

Functional Analysis, Operator Theory, von Neumann Algebra

Pages

v, 85 pages

Bibliography

Includes bibliographical references (pages 78-85).

Copyright

Copyright © 2017 Sujan Pant

Included in

Mathematics Commons

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