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On subdirect products of type FPn of limit groups over Droms RAAGs

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups Connections with homological algebra and category theory

Published online by Cambridge University Press:  11 October 2023

DESSISLAVA H. KOCHLOUKOVA  and
JONE LOPEZ DE GAMIZ ZEARRA
DESSISLAVA H. KOCHLOUKOVA
Affiliation:
Institute of Mathematics, Statistics and Scientific Computing, State University of Campinas, Rua Sèrgio Buarque de Holanda 651, São Paulo, 13083, Brazil. e-mail: desi@ime.unicamp.br
JONE LOPEZ DE GAMIZ ZEARRA
Affiliation:
Department of Mathematics, Vanderbilt University, Stevenson Center Ln 1326, Nashville, 27205, U.S.A. e-mail: jone.lopez.de.gamiz.zearra@vanderbilt.edu
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Abstract

We generalise some known results for limit groups over free groups and residually free groups to limit groups over Droms RAAGs and residually Droms RAAGs, respectively. We show that limit groups over Droms RAAGs are free-by-(torsion-free nilpotent). We prove that if S is a full subdirect product of type $FP_s(\mathbb{Q})$ of limit groups over Droms RAAGs with trivial center, then the projection of S to the direct product of any s of the limit groups over Droms RAAGs has finite index. Moreover, we compute the growth of homology groups and the volume gradients for limit groups over Droms RAAGs in any dimension and for finitely presented residually Droms RAAGs of type $FP_m$ in dimensions up to m. In particular, this gives the values of the analytic $L^2$-Betti numbers of these groups in the respective dimensions.

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20J05: Homological methods in group theory 20E08: Groups acting on trees
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 176 , Issue 2 , March 2024 , pp. 417 - 440
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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