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Preprint Number 2063
2063. Olga Kharlampovich, Alexei Myasnikov, Mahmood Sohrabi
Rich groups, weak second order logic, and applications
Submission date: 27 September 2021
This a mainly expository paper, the final version of which appeared as a contribution to the book "Groups and Model Theory, GAGTA Book 2", edited by Kharlampovich and Sklinos, published in 2021 by de Gruyter Journal-ref: Groups and Model Theory, GAGTA Book 2, edited by Kharlampovich and Sklinos, published in 2021 by de Gruyter.
In this paper we initiate a study of first-order rich groups, i.e., groups where the first-order logic has the same power as the weak second order logic. Surprisingly, there are quite a lot of finitely generated rich groups, they are somewhere in between hyperbolic and nilpotent groups (these ones are not rich). We provide some methods to prove that groups (and other structures) are rich and describe some of their properties. As corollaries we look at Malcev's problems in various groups.
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