Random groups at density 𝑑<3/14 act non-trivially on a CAT(0) cube complex

Montee, MurphyKate

doi:10.1090/tran/8778

Random groups at density $d<3/14$ act non-trivially on a CAT(0) cube complex
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by MurphyKate Montee;

Trans. Amer. Math. Soc. 376 (2023), 1653-1682

DOI: https://doi.org/10.1090/tran/8778

Published electronically: November 4, 2022

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Abstract:

For random groups in the Gromov density model at $d<3/14$, we construct walls in the Cayley complex $X$ which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities $d<1/5$ and $d<5/24$, respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density $d<1/4$.

References

Calum J. Ashcroft and Colva M. Roney-Dougal, On random presentations with fixed relator length, Comm. Algebra 48 (2020), no. 5, 1904–1918. MR 4085767, DOI 10.1080/00927872.2019.1710161
Calum J. Ashcroft, Property (T) in density-type models of random groups, 2021.
François Dahmani, Vincent Guirardel, and Piotr Przytycki, Random groups do not split, Math. Ann. 349 (2011), no. 3, 657–673. MR 2755002, DOI 10.1007/s00208-010-0532-4
Yen Duong, On Random Groups: The Square Model at Density d < 1/3 and as Quotients of Free Nilpotent Groups, ProQuest LLC, Ann Arbor, MI, 2017. Thesis (Ph.D.)–University of Illinois at Chicago. MR 3781851
Étienne Ghys and Pierre de la Harpe, Quasi-isométries et quasi-géodésiques, Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988) Progr. Math., vol. 83, Birkhäuser Boston, Boston, MA, 1990, pp. 79–102 (French). MR 1086653, DOI 10.1007/978-1-4684-9167-8_{5}
M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
Marcin Kotowski and MichałKotowski, Random groups and property $(T)$: Żuk’s theorem revisited, J. Lond. Math. Soc. (2) 88 (2013), no. 2, 396–416. MR 3106728, DOI 10.1112/jlms/jdt024
MurphyKate Montee, Property (T) in $k$-gonal random groups, Glasg. Math. J., 2021.
John M. Mackay and Piotr Przytycki, Balanced walls for random groups, Michigan Math. J. 64 (2015), no. 2, 397–419. MR 3359032, DOI 10.1307/mmj/1434731930
Graham A. Niblo and Martin A. Roller, Groups acting on cubes and Kazhdan’s property (T), Proc. Amer. Math. Soc. 126 (1998), no. 3, 693–699. MR 1459140, DOI 10.1090/S0002-9939-98-04463-3
Tomasz Odrzygóźdź, Nonplanar isoperimetric inequality for random groups, 2014, http://students.mimuw.edu.pl/\~{}to277393/web/files/nonplanar.pdf.
Tomasz Odrzygóźdź, The square model for random groups, Colloq. Math., 142 (2014).
Tomasz Odrzygóźdź, Bent walls for random groups in the square and hexagonal model, 2019.
Y. Ollivier, Sharp phase transition theorems for hyperbolicity of random groups, Geom. Funct. Anal. 14 (2004), no. 3, 595–679. MR 2100673, DOI 10.1007/s00039-004-0470-y
Yann Ollivier, A January 2005 invitation to random groups, Ensaios Matemáticos [Mathematical Surveys], vol. 10, Sociedade Brasileira de Matemática, Rio de Janeiro, 2005. MR 2205306
Yann Ollivier, Some small cancellation properties of random groups, Internat. J. Algebra Comput. 17 (2007), no. 1, 37–51. MR 2300404, DOI 10.1142/S021819670700338X
Yann Ollivier and Daniel T. Wise, Cubulating random groups at density less than $1/6$, Trans. Amer. Math. Soc. 363 (2011), no. 9, 4701–4733. MR 2806688, DOI 10.1090/S0002-9947-2011-05197-4
Michah Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. (3) 71 (1995), no. 3, 585–617. MR 1347406, DOI 10.1112/plms/s3-71.3.585
A. Żuk, Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. 13 (2003), no. 3, 643–670. MR 1995802, DOI 10.1007/s00039-003-0425-8