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Divergence and quasi-isometry classes of random Gromov’s monsters

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  18 February 2021

DOMINIK GRUBER  and
ALESSANDRO SISTO
DOMINIK GRUBER
Affiliation:
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland. e-mail: dom.gruber@gmx.at
ALESSANDRO SISTO
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh e-mail: a.sisto@hw.ac.uk
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Abstract

We show that Gromov’s monsters arising from i.i.d. random labellings of expanders (that we call random Gromov’s monsters) have linear divergence along a subsequence, so that in particular they do not contain Morse quasigeodesics, and they are not quasi-isometric to Gromov’s monsters arising from graphical small cancellation labellings of expanders.

Moreover, by further studying the divergence function, we show that there are uncountably many quasi-isometry classes of random Gromov’s monsters.

MSC classification

Primary: 20F65: Geometric group theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 2 , September 2021 , pp. 249 - 264
Copyright
© Cambridge Philosophical Society 2021

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References

REFERENCES

Arzhantseva, G. and Biswas, A.. Large girth graphs with bounded diameter-by-girth ratio. ArXiv e-prints (March 2018), available at 1803.09229.Google Scholar
Arzhantseva, G. and Delzant, T.. Examples of random groups. Preprint available at http://www.mat.univie.ac.at/arjantseva/Abs/random.pdf (2008).Google Scholar
Björklund, A., Husfeldt, T. and Khanna, S.. Approximating longest directed paths and cycles. Automata, languages and programming (2004), pp. 222–233.CrossRefGoogle Scholar
Bridson, M. R. and Haefliger, A.. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319 (Springer-Verlag, Berlin, 1999).Google Scholar
Coornaert, M., Delzant, T. and Papadopoulos, A.. Géométrie et théorie des groups. Lecture Notes in Mathematics, vol. 1441 (Springer-Verlag, Berlin, 1990).Google Scholar
Coulon, R.. On the geometry of Burnside quotients of torsion free hyperbolic groups. Internat. J. Algebra Comput. 24 (2014), no. 3, 251–345.CrossRefGoogle Scholar
Druţu, C., Mozes, S. and Sapir, M.. Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Amer. Math. Soc. 362 (2010), no. 5, 2451–2505.CrossRefGoogle Scholar
Finn-Sell, M.. Almost quasi-isometries and more non-C*-exact groups. Math. Proc. Camb. Philos. Soc. 162 (2017), no. 3, 393–403.CrossRefGoogle Scholar
Gersten, S. M.. Quadratic divergence of geodesics in CAT(0) spaces. Geom. Funct. Anal. 4 (1994), no. 1, 37–51.CrossRefGoogle Scholar
Gromov, M.. Asymptotic invariants of infinite groups. London Math. Soc. Lecture Note Ser., vol. 182 (Cambridge Univ. Press, Cambridge, 1993).Google Scholar
Gromov, M.. Random walk in random groups. Geom. Funct. Anal. 13 (2003), no. 1, 73–146.CrossRefGoogle Scholar
Gromov, Misha. Spaces and questions. Geom. Funct. Anal. Special Volume, Part I (2000), 118–161. GAFA 2000 (Tel Aviv, 1999).Google Scholar
Gruber, D.. Groups with graphical C(6) and C(7) small cancellation presentations. Trans. Amer. Math. Soc. 367 (2015), no. 3, 2051–2078.CrossRefGoogle Scholar
Gruber, D. and Sisto, A.. Infinitely presented graphical small cancellation groups are acylindrically hyperbolic. to appear in Ann. Inst. Fourier (2018).Google Scholar
Gruber, D., Sisto, A. and Tessera, R.. Gromov’s random monsters do not act non-elementarily on hyperbolic spaces. Proc. Amer. Math. Soc. 148 (2020), 2773–2782.CrossRefGoogle Scholar
Higson, N., Lafforgue, V. and Skandalis, G.. Counterexamples to the Baum-Connes conjecture. Geom. Funct. Anal. 12 (2002), no. 2, 330–354.CrossRefGoogle Scholar
Hume, D.. A continuum of expanders. Fund. Math. 238 (2017), no. 2, 143–152.CrossRefGoogle Scholar
Kesten, H.. Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 (1959), 336–354.CrossRefGoogle Scholar
Kowalski, E.. An introduction to expander graphs. Lecture notes available at http://www.math.ethz.ch/kowalski/expander-graphs.pdf (2015).Google Scholar
Lubotzky, A.. Discrete groups, expanding graphs and invariant measures. Modern Birkhäuser Classics (Birkhäuser Verlag, Basel, 2010). With an appendix by Jonathan D. Rogawski, Reprint of the 1994 edition.Google Scholar
Margulis, G. A.. Explicit constructions of graphs without short cycles and low density codes. Combinatorica 2 (1982), no. 1, 71–78.CrossRefGoogle Scholar
Matoušek, Jiří. On embedding expanders into lp spaces. Israel J. Math. 102 (1997), 189–197.CrossRefGoogle Scholar
Ollivier, Y.. On a small cancellation theorem of Gromov. Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 1, 75–89.CrossRefGoogle Scholar
Yu, A.. Osin, Ol’shanskii, D. V. and Sapir, M. V.. Lacunary hyperbolic groups. Geom. Topol. 13 (2009), no. 4, 2051–2140. With an appendix by Michael Kapovich and Bruce Kleiner.Google Scholar
Osajda, D.. Small cancellation labellings of some infinite graphs and applications. Acta Math. 225 (2020), no. 1, 159–191.CrossRefGoogle Scholar
Selberg, A.. On the estimation of Fourier coefficients of modular forms. Proc. Sympos. Pure Math., Vol. VIII, 1965, pp. 1–15.Google Scholar
Serre, J.-P.. Trees, Springer Monographs in Mathematics (Springer-Verlag, Berlin, 2003). Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation.Google Scholar
Sisto, A.. Quasi-convexity of hyperbolically embedded subgroups. Math. Z. 283 (2016), no. 3-4, 649–658.CrossRefGoogle Scholar
Woess, W.. Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, vol. 138 (Cambridge University Press, Cambridge).Google Scholar