Proper 2–equivalences between infinite ended finitely presented groups

Cardenas, Manuel; Fernández Lasheras, Francisco; Quintero, Antonio; Roy, Ranja

doi:10.2140/agt.2023.23.1

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Proper $2$–equivalences between infinite ended finitely presented groups

Manuel Cárdenas, Francisco Fernández Lasheras, Antonio Quintero and Ranja Roy

Algebraic & Geometric Topology 23 (2023) 1–11

DOI: 10.2140/agt.2023.23.1

Abstract

Recall that two finitely presented groups $G$ and $H$ are “proper $2$ –equivalent” if they can be realized by finite $2$ –dimensional CW–complexes whose universal covers are proper $2$ –equivalent as (strongly) locally finite CW–complexes. This purely topological relation is coarser than the quasi-isometry relation, and those groups which are $1$ –ended and semistable at infinity are classified, up to proper $2$ –equivalence, by their fundamental pro-group. We show that if $G$ and $H$ are proper $2$ –equivalent and semistable at each end, then any two finite graph of groups decompositions of $G$ and $H$ with finite edge groups and finitely presented vertex groups with at most one end must have the same set of proper $2$ –equivalence classes of (infinite) nonsimply connected at infinity vertex groups (without multiplicities). Moreover, those simply connected at infinity vertex groups in such a decomposition (if any) are all proper $2$ –equivalent to $ℤ \times ℤ \times ℤ$ . Thus, under the semistability hypothesis, this answers a question concerning the classification of infinite ended finitely presented groups up to proper $2$ –equivalence, and shows again the behavior of proper $2$ –equivalences versus quasi-isometries, in which the geometry of the group is taken into account.

Keywords

proper homotopy, proper $2$–equivalence, quasi-isometry, finitely presented group

Mathematical Subject Classification 2010

Primary: 57M07

Secondary: 57M10

References

Publication

Received: 29 January 2020

Revised: 10 August 2021

Accepted: 5 October 2021

Published: 27 March 2023

Authors

Manuel Cárdenas
Departamento de Geometría y Topología Facultad Matematicas Universidad de Sevilla Seville Spain

Francisco Fernández Lasheras
Departamento de Geometría y Topología Facultad Matematicas Universidad de Sevilla Seville Spain

Antonio Quintero
Departamento de Geometría y Topología Facultad Matematicas Universidad de Sevilla Seville Spain

Ranja Roy
College of Arts and Sciences New York Institute of Technology Old Westbury NY United States

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Volume 23, issue 1 (2023)