## Duke Mathematical Journal

### On the cubical geometry of Higman’s group

Alexandre Martin

#### Abstract

We investigate the cocompact action of Higman’s group on a $\operatorname{CAT}(0)$ square complex associated to its standard presentation. We show that this action is in a sense intrinsic, which allows for the use of geometric techniques to study the endomorphisms of the group, and we show striking similarities with mapping class groups of hyperbolic surfaces, outer automorphism groups of free groups, and linear groups over the integers. We compute explicitly the automorphism group and outer automorphism group of Higman’s group and show that the group is both Hopfian and co-Hopfian. We actually prove a stronger rigidity result about the endomorphisms of Higman’s group: every nontrivial morphism from the group to itself is an automorphism. We also study the geometry of the action and prove a surprising result: although the $\operatorname{CAT}(0)$ square complex acted upon contains uncountably many flats, the Higman group does not contain subgroups isomorphic to $\mathbb{Z}^{2}$. Finally, we show that this action possesses features reminiscent of negative curvature, which we use to prove a refined version of the Tits alternative for Higman’s group.

#### Article information

Source
Duke Math. J., Volume 166, Number 4 (2017), 707-738.

Dates
Received: 19 August 2015
Revised: 26 May 2016
First available in Project Euclid: 3 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1483412430

Digital Object Identifier
doi:10.1215/00127094-3715913

Mathematical Reviews number (MathSciNet)
MR3619304

Zentralblatt MATH identifier
06706844

#### Citation

Martin, Alexandre. On the cubical geometry of Higman’s group. Duke Math. J. 166 (2017), no. 4, 707--738. doi:10.1215/00127094-3715913. https://projecteuclid.org/euclid.dmj/1483412430

#### References

• [1] G. C. Bell and A. N. Dranishnikov, A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory, Trans. Amer. Math. Soc. 358 (2006), 4749–4764.
• [2] M. Bestvina, M. Feighn, and M. Handel, The Tits alternative for $\operatorname{Out}(F_{n})$, I: Dynamics of exponentially-growing automorphisms, Ann. of Math. (2) 151 (2000), 517–623.
• [3] M. Bestvina, M. Feighn, and M. Handel, The Tits alternative for $\operatorname{Out}(F_{n})$, II: A Kolchin type theorem, Ann. of Math. (2) 161 (2005), 1–59.
• [4] M. R. Bridson, On the semisimplicity of polyhedral isometries, Proc. Amer. Math. Soc. 127 (1999), 2143–2146.
• [5] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wissen. 319, Springer, Berlin, 1999.
• [6] M. R. Bridson and K. Vogtmann, Automorphisms of automorphism groups of free groups, J. Algebra 229 (2000), 785–792.
• [7] P.-E. Caprace and M. Sageev, Rank rigidity for $\operatorname{CAT}(0)$ cube complexes, Geom. Funct. Anal. 21 (2011), 851–891.
• [8] R. Charney and M. Adavis, When is a Coxeter system determined by its Coxeter group? J. Lond. Math. Soc. (2) 61 (2000), 441–461.
• [9] V. N. Gerasimov, “Semi-splittings of groups and actions on cubings” in Algebra, Geometry, Analysis and Mathematical Physics (Novosibirsk, 1996) (in Russian), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat. 190, Novosibirsk, 1997, 91–109.
• [10] W. Gottschalk, “Some general dynamical notions” in Recent Advances in Topological Dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972), Lecture Notes in Math. 318, Springer, Berlin, 1973, 120–125.
• [11] M. Gromov, “Asymptotic invariants of infinite groups” in Geometric Group Theory, Vol. 2 (Sussex, 1991), Lond. Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press, Cambridge, 1993, 1–295.
• [12] M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS) 1 (1999), 109–197.
• [13] E. K. Grossman, On the residual finiteness of certain mapping class groups, J. Lond. Math. Soc. 9 (1974/1975), 160–164.
• [14] F. Haglund and D. T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), 1551–1620.
• [15] G. Higman, A finitely generated infinite simple group, J. Lond. Math. Soc. 26 (1951), 61–64.
• [16] L. K. Hua and I. Reiner, Automorphisms of the unimodular group, Trans. Amer. Math. Soc. 71 (1951), 331–348.
• [17] N. V. Ivanov, Algebraic properties of the Teichmüller modular group, Dokl. Akad. Nauk SSSR 275 (1984), 786–789.
• [18] N. V. Ivanov and J. D. McCarthy, On injective homomorphisms between Teichmüller modular groups, I, Invent. Math. 135 (1999), 425–486.
• [19] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Classics in Math., Springer, Berlin, 2001.
• [20] A. MalcEV, On isomorphic matrix representations of infinite groups, Rec. Math. [Mat. Sbornik] N.S. 8 (1940), no. 50, 405–422.
• [21] A. Martin, Acylindrical actions on $\operatorname{CAT}(0)$ square complexes, preprint, arXiv:1509.03131.
• [22] J. P. McCammond and D. T. Wise, Fans and ladders in small cancellation theory, Proc. Lond. Math. Soc. (3) 84 (2002), 599–644.
• [23] J. D. McCarthy, A “Tits-alternative” for subgroups of surface mapping class groups, Trans. Amer. Math. Soc. 291 (1985), 583–612.
• [24] J. D. McCarthy, Automorphisms of surface mapping class groups: A recent theorem of N. Ivanov, Invent. Math. 84 (1986), 49–71.
• [25] A. Minasyan and D. Osin, Acylindrically hyperbolic groups acting on trees, Math. Ann. 362 (2015), 1055–1105.
• [26] D. Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), 851–888.
• [27] V. P. Platonov and O. I. Tavgen, On the Grothendieck problem of profinite completions of groups, Dokl. Akad. Nauk SSSR 288 (1986), 1054–1058.
• [28] G. Prasad, Discrete subgroups isomorphic to lattices in semisimple Lie groups, Amer. J. Math. 98 (1976), 241–261.
• [29] M. Sageev and D. T. Wise, The Tits alternative for $\operatorname{CAT}(0)$ cubical complexes, Bull. Lond. Math. Soc. 37 (2005), 706–710.
• [30] P. E. Schupp, Small cancellation theory over free products with amalgamation, Math. Ann. 193 (1971), 255–264.
• [31] Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank $1$ Lie groups, II, Geom. Funct. Anal. 7 (1997), 561–593.
• [32] Z. Sela, Endomorphisms of hyperbolic groups, I: The Hopf property, Topology 38 (1999), 301–321.
• [33] J.-P. Serre, Arbres, amalgames, $\mathrm{SL}_{2}$, Astérisque 46, Soc. Math. France, Paris, 1977.
• [34] J. R. Stallings, “Non-positively curved triangles of groups” in Group Theory from a Geometrical Viewpoint (Trieste, 1990), World Sci. Publ., River Edge, N.J., 1991, 491–503.
• [35] J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270.
• [36] G. Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2) 147 (1998), 325–355.
• [37] G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), 201–240.