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Folding-like techniques for CAT(0) cube complexes

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  28 October 2021

MICHAEL BEN–ZVI ,
ROBERT KROPHOLLER  and
RYLEE ALANZA LYMAN
MICHAEL BEN–ZVI
Affiliation:
Colby College, Department of Mathematics Davis Building, Second Floor 5830 Mayflower Hill Waterville, Maine 04901 U.S.A. e-mail: mbenzvi@colby.edu
ROBERT KROPHOLLER
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL. e-mail: robertkropholler@gmail.com
RYLEE ALANZA LYMAN
Affiliation:
Department of Mathematics and Computer Science, Rutgers University–Newark Smith Hall, Room 216, 101 Warren Street Newark, NJ 072102, U.S.A. e-mail: ryleealanza@gmail.com
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Abstract

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 1 , July 2022 , pp. 227 - 238
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 –390685587, Mathematics Münster: Dynamics Geometry Structure.

References

B., Beeker and N., Lazarovich. Stallings’ folds for cube complexes. Israel J. Math. 227(1):331–363, (2018).Google Scholar
Bridson, R. and AndrÉ, Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (Springer-Verlag, Berlin, 1999).CrossRefGoogle Scholar
S. Brown. Geometric structures on negatively curved groups and their subgroups. PhD. thesis University College London (2016).Google Scholar
P.-E., Caprace and M., Sageev. Rank rigidity for CAT(0) cube complexes. Geom. Funct. Anal. 21(4): (2011) 851–891.Google Scholar
Pallavi, Dani and I., Levcovitz. Subgroups of right-angled coxeter groups via stallings-like techniques. Available at arXiv:1908.09046 [Math.GT] (2019).Google Scholar
Gromov, M.. Hyperbolic groups. In Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ. (Springer, New York, 1987), pages 75–263.CrossRefGoogle Scholar
F., Haglund. Finite index subgroups of graph products. Geom. Dedicata 135: (2008), 167–209.Google Scholar
O., Kharlampovich, A., Miasnikov, and P., Weil. Stallings graphs for quasi-convex subgroups. J. Algebra 488: (2017), 442–483.Google Scholar
M., Roller. Poc sets, median algebras and group actions. habilitation thesis. Available at arXiv:1607.07747 [math.GN] (July 2016).Google Scholar
M., Sageev. Ends of group pairs and non-positively curved cube complexes. Proc. London Math. Soc. (3), 71(3): (1995), 585–617.CrossRefGoogle Scholar
M., Sageev. $\textrm{CAT(0)}$ cube complexes and groups. In Geometric group theory, volume 21 of IAS/Park City Math. Ser., pages 7–54 (Amer. Math. Soc., Providence, RI, 2014).CrossRefGoogle Scholar
Stallings, J. R.. Topology of finite graphs. Invent. Math. 71(3): (1983), 551–565.CrossRefGoogle Scholar