Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

A Cantor manifold C is a non-compact manifold obtained by gluing (holed) copies of a fixed compact manifold Y in a tree-like manner. Generalizing classical families of groups due to Brin [10], Dehornoy [18] and Funar–Kapoudjian [21, 23, 1], we introduce the asymptotic mapping class group B of C, whose elements are proper isotopy classes of selfdiffeomorphisms of C that are eventually trivial. The group B happens to be an extension of a Higman-Thompson group by a direct limit of mapping class groups of compact submanifolds of C. We construct an infinite-dimensional contractible cube complex X on which B acts. For certain well-studied families of manifolds, we prove that B is of type F8 and that X is CATp0q; more concretely, our methods apply for example when Y is diffeomorphic to S 1 ˆS 1 , S 2 ˆS 1 , or S n ˆS n for n ě 3. In these cases, B contains, respectively, the mapping class group of every compact surface with boundary; the automorphism group of the free group on k generators for all k; and an infinite family of (arithmetic) symplectic or orthogonal groups. In particular, if Y – S 2 or S 1 ˆ S 1 , our result gives a positive answer to [24, Problem 3] and [3, Question 5.32]. In addition, for Y – S 1 ˆ S 1 or S 2 ˆ S 1 , the homology of B coincides with the stable homology of the relevant mapping class groups, as studied by Harer [31] and Hatcher– Wahl [39].

Aramayona, Javier

3291de41-c6c0-465a-9af5-7bbafef74784

Bux, Kai-Uwe

f54c6b1e-9e0a-4909-a3ad-8466ed9f00f0

Flechsig, Jonas

68159729-d25a-4e06-bb44-f896c4e04809

Petrosyan, Nansen

f169cfd6-aeee-4ad2-b147-0bf77dd1f9b6

Wu, Xiaolei

d85130da-ae18-45d7-a8d3-dec38230c76c

Aramayona, Javier

3291de41-c6c0-465a-9af5-7bbafef74784

Bux, Kai-Uwe

f54c6b1e-9e0a-4909-a3ad-8466ed9f00f0

Flechsig, Jonas

68159729-d25a-4e06-bb44-f896c4e04809

Petrosyan, Nansen

f169cfd6-aeee-4ad2-b147-0bf77dd1f9b6

Wu, Xiaolei

d85130da-ae18-45d7-a8d3-dec38230c76c

Aramayona, Javier, Bux, Kai-Uwe, Flechsig, Jonas, Petrosyan, Nansen and Wu, Xiaolei
(2024)
Asymptotic mapping class groups of Cantor manifolds and their finiteness properties.
*Revista Matematica Iberoamericana*.
(In Press)

## Abstract

A Cantor manifold C is a non-compact manifold obtained by gluing (holed) copies of a fixed compact manifold Y in a tree-like manner. Generalizing classical families of groups due to Brin [10], Dehornoy [18] and Funar–Kapoudjian [21, 23, 1], we introduce the asymptotic mapping class group B of C, whose elements are proper isotopy classes of selfdiffeomorphisms of C that are eventually trivial. The group B happens to be an extension of a Higman-Thompson group by a direct limit of mapping class groups of compact submanifolds of C. We construct an infinite-dimensional contractible cube complex X on which B acts. For certain well-studied families of manifolds, we prove that B is of type F8 and that X is CATp0q; more concretely, our methods apply for example when Y is diffeomorphic to S 1 ˆS 1 , S 2 ˆS 1 , or S n ˆS n for n ě 3. In these cases, B contains, respectively, the mapping class group of every compact surface with boundary; the automorphism group of the free group on k generators for all k; and an infinite family of (arithmetic) symplectic or orthogonal groups. In particular, if Y – S 2 or S 1 ˆ S 1 , our result gives a positive answer to [24, Problem 3] and [3, Question 5.32]. In addition, for Y – S 1 ˆ S 1 or S 2 ˆ S 1 , the homology of B coincides with the stable homology of the relevant mapping class groups, as studied by Harer [31] and Hatcher– Wahl [39].

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** 2110.05318
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## More information

Accepted/In Press date: 8 July 2024

## Identifiers

Local EPrints ID: 452481

URI: http://eprints.soton.ac.uk/id/eprint/452481

ISSN: 0213-2230

PURE UUID: ff166460-581b-4c80-a3fc-8d07fda3354f

## Catalogue record

Date deposited: 11 Dec 2021 11:16

Last modified: 11 Sep 2024 01:59

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## Contributors

Author:
Javier Aramayona

Author:
Kai-Uwe Bux

Author:
Jonas Flechsig

Author:
Xiaolei Wu

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