On the first-order genus of wreath products and their central extensions

Olga Kharlampovich; Alexei Miasnikov; Denis Osin

doi:10.1016/j.jalgebra.2026.06.002

On the first-order genus of wreath products and their central extensions

Olga Kharlampovich
, Alexei Miasnikov
, Denis Osin

Department of Mathematical Sciences

City University of New York
Vanderbilt University

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that groups of the form Z^mwrZⁿ, where m,n∈N, are regularly bi-interpretable with Z and therefore are first-order rigid: every finitely generated group elementarily equivalent to Z^mwrZⁿ is isomorphic to Z^mwrZⁿ. On the other hand, we show that Z²wrZ admits 2^ℵ₀ elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel.

Original language	English
Pages (from-to)	256-292
Number of pages	37
Journal	Journal of Algebra
Volume	707
DOIs	https://doi.org/10.1016/j.jalgebra.2026.06.002
State	Published - 1 Dec 2026

Keywords

First-order genus
Metabelian groups
Space of marked groups

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10.1016/j.jalgebra.2026.06.002

Cite this

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title = "On the first-order genus of wreath products and their central extensions",

abstract = "We prove that groups of the form ZmwrZn, where m,n∈N, are regularly bi-interpretable with Z and therefore are first-order rigid: every finitely generated group elementarily equivalent to ZmwrZn is isomorphic to ZmwrZn. On the other hand, we show that Z2wrZ admits 2ℵ0 elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel.",

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author = "Olga Kharlampovich and Alexei Miasnikov and Denis Osin",

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AU - Kharlampovich, Olga

AU - Miasnikov, Alexei

AU - Osin, Denis

PY - 2026/12/1

Y1 - 2026/12/1

N2 - We prove that groups of the form ZmwrZn, where m,n∈N, are regularly bi-interpretable with Z and therefore are first-order rigid: every finitely generated group elementarily equivalent to ZmwrZn is isomorphic to ZmwrZn. On the other hand, we show that Z2wrZ admits 2ℵ0 elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel.

AB - We prove that groups of the form ZmwrZn, where m,n∈N, are regularly bi-interpretable with Z and therefore are first-order rigid: every finitely generated group elementarily equivalent to ZmwrZn is isomorphic to ZmwrZn. On the other hand, we show that Z2wrZ admits 2ℵ0 elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel.

KW - First-order genus

KW - Metabelian groups

KW - Space of marked groups

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UR - https://www.scopus.com/pages/publications/105041345477#tab=citedBy

U2 - 10.1016/j.jalgebra.2026.06.002

DO - 10.1016/j.jalgebra.2026.06.002

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SN - 0021-8693

VL - 707

SP - 256

EP - 292

JO - Journal of Algebra

JF - Journal of Algebra

ER -

On the first-order genus of wreath products and their central extensions

Abstract

Keywords

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