Groups of proper homotopy equivalences of graphs and Nielsen Realization

Yael Algom-Kfir; Mladen Bestvina

doi:10.1090/conm/812/16259

Groups of proper homotopy equivalences of graphs and Nielsen Realization

Yael Algom-Kfir, Mladen Bestvina

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

For a locally finite connected graph X we consider the group M aps(X) of proper homotopy equivalences of X. We show that it has a natural Polish group topology, and we propose these groups as an analog of big mapping class groups. We prove the Nielsen Realization theorem: if H is a compact subgroup of Maps(X) then X is proper homotopy equivalent to a graph Y so that H is realized by simplicial isomorphisms of Y.

Original language	English
Title of host publication	Topology at Infinity of Discrete Groups - AMS Special Session on Ends and Boundaries of Groups In honor of Michael Mihalik’s 70th Birthday, 2023
Editors	Ross Geoghegan, Craig R. Guilbault, Kim Ruane
Publisher	American Mathematical Society
Pages	1-31
Number of pages	31
ISBN (Print)	9781470478636
DOIs	https://doi.org/10.1090/conm/812/16259
State	Published - 2025
Event	AMS Special Session on Ends and Boundaries of Groups In honor of Michael Mihalik’s 70th Birthday, 2023 - Cincinnati, United States Duration: 15 Apr 2023 → 16 Apr 2023

Publication series

Name	Contemporary Mathematics
Volume	812
ISSN (Print)	0271-4132
ISSN (Electronic)	1098-3627

Conference

Conference	AMS Special Session on Ends and Boundaries of Groups In honor of Michael Mihalik’s 70th Birthday, 2023
Country/Territory	United States
City	Cincinnati
Period	15/04/23 → 16/04/23

Bibliographical note

Publisher Copyright:
© 2025 American Mathematical Society.

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1090/conm/812/16259

Cite this

Algom-Kfir, Y., & Bestvina, M. (2025). Groups of proper homotopy equivalences of graphs and Nielsen Realization. In R. Geoghegan, C. R. Guilbault, & K. Ruane (Eds.), Topology at Infinity of Discrete Groups - AMS Special Session on Ends and Boundaries of Groups In honor of Michael Mihalik’s 70th Birthday, 2023 (pp. 1-31). (Contemporary Mathematics; Vol. 812). American Mathematical Society. https://doi.org/10.1090/conm/812/16259

Algom-Kfir, Yael ; Bestvina, Mladen. / Groups of proper homotopy equivalences of graphs and Nielsen Realization. Topology at Infinity of Discrete Groups - AMS Special Session on Ends and Boundaries of Groups In honor of Michael Mihalik’s 70th Birthday, 2023. editor / Ross Geoghegan ; Craig R. Guilbault ; Kim Ruane. American Mathematical Society, 2025. pp. 1-31 (Contemporary Mathematics).

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abstract = "For a locally finite connected graph X we consider the group M aps(X) of proper homotopy equivalences of X. We show that it has a natural Polish group topology, and we propose these groups as an analog of big mapping class groups. We prove the Nielsen Realization theorem: if H is a compact subgroup of Maps(X) then X is proper homotopy equivalent to a graph Y so that H is realized by simplicial isomorphisms of Y.",

author = "Yael Algom-Kfir and Mladen Bestvina",

note = "Publisher Copyright: {\textcopyright} 2025 American Mathematical Society.; AMS Special Session on Ends and Boundaries of Groups In honor of Michael Mihalik{\textquoteright}s 70th Birthday, 2023 ; Conference date: 15-04-2023 Through 16-04-2023",

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doi = "10.1090/conm/812/16259",

language = "English",

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Algom-Kfir, Y & Bestvina, M 2025, Groups of proper homotopy equivalences of graphs and Nielsen Realization. in R Geoghegan, CR Guilbault & K Ruane (eds), Topology at Infinity of Discrete Groups - AMS Special Session on Ends and Boundaries of Groups In honor of Michael Mihalik’s 70th Birthday, 2023. Contemporary Mathematics, vol. 812, American Mathematical Society, pp. 1-31, AMS Special Session on Ends and Boundaries of Groups In honor of Michael Mihalik’s 70th Birthday, 2023, Cincinnati, United States, 15/04/23. https://doi.org/10.1090/conm/812/16259

Groups of proper homotopy equivalences of graphs and Nielsen Realization. / Algom-Kfir, Yael; Bestvina, Mladen.
Topology at Infinity of Discrete Groups - AMS Special Session on Ends and Boundaries of Groups In honor of Michael Mihalik’s 70th Birthday, 2023. ed. / Ross Geoghegan; Craig R. Guilbault; Kim Ruane. American Mathematical Society, 2025. p. 1-31 (Contemporary Mathematics; Vol. 812).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Algom-Kfir Y, Bestvina M. Groups of proper homotopy equivalences of graphs and Nielsen Realization. In Geoghegan R, Guilbault CR, Ruane K, editors, Topology at Infinity of Discrete Groups - AMS Special Session on Ends and Boundaries of Groups In honor of Michael Mihalik’s 70th Birthday, 2023. American Mathematical Society. 2025. p. 1-31. (Contemporary Mathematics). doi: 10.1090/conm/812/16259

Groups of proper homotopy equivalences of graphs and Nielsen Realization

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