Commensurating HNN extensions: nonpositive curvature and biautomaticity

Leary, Ian J¹ ; Minasyan, Ashot¹

¹ School of Mathematical Sciences, University of Southampton, Southampton, United Kingdom

Geometry & topology, Tome 25 (2021) no. 4, pp. 1819-1860.

Résumé

We show that the commensurator of any quasiconvex abelian subgroup in a biautomatic group is small, in the sense that it has finite image in the abstract commensurator of the subgroup. Using this criterion we exhibit groups that are $CAT\left(0\right)$ but not biautomatic. These groups also resolve a number of other questions concerning $CAT\left(0\right)$ groups.

Détail

DOI : 10.2140/gt.2021.25.1819

Classification : 20F10, 20F67, 20E06
Keywords: commensurating HNN extension, biautomatic group, CAT(0) group

Bibliographie

[1] J M Alonso, M R Bridson, Semihyperbolic groups, Proc. London Math. Soc. 70 (1995) 56 | DOI

[2] A Amrhein, Characterizing biautomatic groups, preprint (2021)

[3] S Andreadakis, E Raptis, D Varsos, Residual finiteness and Hopficity of certain HNN extensions, Arch. Math. Basel 47 (1986) 1 | DOI

[4] S Andreadakis, E Raptis, D Varsos, Hopficity of certain HNN-extensions, Comm. Algebra 20 (1992) 1511 | DOI

[5] G N Arzhantseva, J Burillo, M Lustig, L Reeves, H Short, E Ventura, Uniform non-amenability, Adv. Math. 197 (2005) 499 | DOI

[6] G Baumslag, S M Gersten, M Shapiro, H Short, Automatic groups and amalgams, J. Pure Appl. Algebra 76 (1991) 229 | DOI

[7] G Baumslag, D Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 (1962) 199 | DOI

[8] M Bestvina, Questions in geometric group theory, (2000)

[9] M R Bridson, Regular combings, nonpositive curvature and the quasiconvexity of abelian subgroups, J. Pure Appl. Algebra 88 (1993) 23 | DOI

[10] M R Bridson, On the growth of groups and automorphisms, Internat. J. Algebra Comput. 15 (2005) 869 | DOI

[11] M R Bridson, R H Gilman, Formal language theory and the geometry of 3–manifolds, Comment. Math. Helv. 71 (1996) 525 | DOI

[12] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999) | DOI

[13] P E Caprace, N Monod, Isometry groups of non-positively curved spaces: discrete subgroups, J. Topol. 2 (2009) 701 | DOI

[14] P E Caprace, N Monod, Erratum and addenda to “Isometry groups of non-positively curved spaces : discrete subgroups”, preprint (2019)

[15] M Casals-Ruiz, I Kazachkov, A Zakharov, Commensurability of Baumslag–Solitar groups, preprint (2019)

[16] M Elder, Automaticity, almost convexity and falsification by fellow traveler properties of some finitely generated groups, PhD thesis, The University of Melbourne (2000)

[17] D B A Epstein, J W Cannon, D F Holt, S V F Levy, M S Paterson, W P Thurston, Word processing in groups, Jones and Bartlett (1992)

[18] D B A Epstein, D F Holt, S E Rees, The use of Knuth–Bendix methods to solve the word problem in automatic groups, J. Symbolic Comput. 12 (1991) 397 | DOI

[19] B Farb, C Hruska, A Thomas, Problems on automorphism groups of nonpositively curved polyhedral complexes and their lattices, from: "Geometry, rigidity, and group actions" (editors B Farb, D Fisher), Univ. Chicago Press (2011) 515 | DOI

[20] S M Gersten, H B Short, Rational subgroups of biautomatic groups, Ann. of Math. 134 (1991) 125 | DOI

[21] J R J Groves, S M Hermiller, Isoperimetric inequalities for soluble groups, Geom. Dedicata 88 (2001) 239 | DOI

[22] J Huang, T Prytuła, Commensurators of abelian subgroups in CAT(0) groups, Math. Z. 296 (2020) 79 | DOI

[23] P H Kropholler, Baumslag–Solitar groups and some other groups of cohomological dimension two, Comment. Math. Helv. 65 (1990) 547 | DOI

[24] R C Lyndon, P E Schupp, Combinatorial group theory, 89, Springer (1977) | DOI

[25] S Macnbsp ;Lane, G Birkhoff, Algebra, Macmillan (1979)

[26] A Maltsev, On isomorphic matrix representations of infinite groups, Rec. Math. [Mat. Sbornik] N.S. 8 (50) (1940) 405

[27] J Mccammond, Algorithms, Dehn functions and automatic groups, preprint (2007)

[28] D Meier, Non-Hopfian groups, J. London Math. Soc. 26 (1982) 265 | DOI

[29] A Minasyan, On conjugacy separability of fibre products, Proc. Lond. Math. Soc. 115 (2017) 1170 | DOI

[30] A Minasyan, Virtual retraction properties in groups, Int. Math. Res. Not. (2019) | DOI

[31] W D Neumann, M Shapiro, Equivalent automatic structures and their boundaries, Internat. J. Algebra Comput. 2 (1992) 443 | DOI

[32] G A Niblo, L D Reeves, The geometry of cube complexes and the complexity of their fundamental groups, Topology 37 (1998) 621 | DOI

[33] D Osajda, T Prytuła, Classifying spaces for families of subgroups for systolic groups, Groups Geom. Dyn. 12 (2018) 1005 | DOI

[34] D V Osin, Kazhdan constants of hyperbolic groups, Funktsional. Anal. i Prilozhen. 36 (2002) 46

[35] T Prytuła, Bredon cohomological dimension for virtually abelian stabilisers for CAT(0) groups, J. Topol. Anal. (2019) | DOI

[36] J P Serre, Trees, Springer (2003) | DOI

[37] T G Group, GAP: Groups, algorithms and programming (2018)

[38] K Whyte, The large scale geometry of the higher Baumslag–Solitar groups, Geom. Funct. Anal. 11 (2001) 1327 | DOI

[39] D T Wise, A non-Hopfian automatic group, J. Algebra 180 (1996) 845 | DOI

[40] D T Wise, Non-positively curved squared complexes : aperiodic tilings and non-residually finite groups, PhD thesis, Princeton University (1996)