Sofic groups

Date

2025-8-1

Author

Yumak, Hakkı Kağan

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Sofic groups, introduced by Gromov, are groups that can be approximated by symmetric groups in a metric sense. They generalize both amenable and residually finite groups. This property has gained significant importance since various famous conjectures hold for sofic groups. Also, interestingly, a major open question is whether all groups are sofic. This thesis is devoted to a survey of sofic groups and related concepts. We first delve into algebraic approximations as a motivation to sofic groups. Then, after defining soficity, we investigate fundamental properties of sofic groups. During this, we explore various characterizations and relationships of soficity to different areas of mathematics. A particular emphasis is placed on universal sofic groups and their properties. Next, it is observed that various important conjectures hold for sofic groups highlighting the importance of this notion. Lastly, stability of groups will be discussed, offering a new approach to answering the fundamental question whether all groups are sofic.

Subject Keywords

Sofic groups, Approximations of groups, Ultraproducts, Stability

URI

https://hdl.handle.net/11511/115509

Collections

Graduate School of Natural and Applied Sciences, Thesis