Strong accessibility for finitely presented groups

Louder, Larsen¹ ; Touikan, Nicholas²

¹ Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom
² Department of Mathematical Sciences, Stevens Institute of Technology, 1 Castle Point on Hudson, Hoboken, NJ 07030, United States

Geometry & topology, Tome 21 (2017) no. 3, pp. 1805-1835.

Résumé

A hierarchy of a group is a rooted tree of groups obtained by iteratively passing to vertex groups of graphs of groups decompositions. We define a (relative) slender JSJ hierarchy for (almost) finitely presented groups and show that it is finite, provided the group in question doesn’t contain any slender subgroups with infinite dihedral quotients and satisfies an ascending chain condition on certain chains of subgroups of edge groups.

As a corollary, slender JSJ hierarchies of finitely presented subgroups of ${SL}_n\left(\mathbb{Z}\right)$ or of hyperbolic groups which are (virtually) without $2$–torsion are finite.

Détail

DOI : 10.2140/gt.2017.21.1805

Classification : 20E08, 20F65, 20F67, 57M60
Keywords: strong accessibility, graph of groups, hierarchy

Bibliographie

[1] M Bestvina, M Feighn, Bounding the complexity of simplicial group actions on trees, Invent. Math. 103 (1991) 449 | DOI

[2] M Bestvina, M Feighn, A counterexample to generalized accessibility, from: "Arboreal group theory" (editor R C Alperin), Math. Sci. Res. Inst. Publ. 19, Springer (1991) 133 | DOI

[3] O V Bogopol’Skiĭ, V N Gerasimov, Finite subgroups of hyperbolic groups, Algebra i Logika 34 (1995) 619 | DOI

[4] B H Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012) 1250016, 1 | DOI

[5] T Delzant, L Potyagailo, Accessibilité hiérarchique des groupes de présentation finie, Topology 40 (2001) 617 | DOI

[6] M J Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985) 449 | DOI

[7] M J Dunwoody, An inaccessible group, from: "Geometric group theory, Vol 1" (editors G A Niblo, M A Roller), London Math. Soc. Lecture Note Ser. 181, Cambridge Univ. Press (1993) 75 | DOI

[8] M J Dunwoody, M E Sageev, JSJ-splittings for finitely presented groups over slender groups, Invent. Math. 135 (1999) 25 | DOI

[9] K Fujiwara, P Papasoglu, JSJ-decompositions of finitely presented groups and complexes of groups, Geom. Funct. Anal. 16 (2006) 70 | DOI

[10] M Gromov, Hyperbolic groups, from: "Essays in group theory" (editor S M Gersten), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 | DOI

[11] D Groves, J F Manning, Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008) 317 | DOI

[12] V Guirardel, Cœur et nombre d’intersection pour les actions de groupes sur les arbres, Ann. Sci. École Norm. Sup. 38 (2005) 847 | DOI

[13] G C Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010) 1807 | DOI

[14] I Kapovich, R Weidmann, On the structure of two-generated hyperbolic groups, Math. Z. 231 (1999) 783 | DOI

[15] O Kharlampovich, A Myasnikov, Irreducible affine varieties over a free group, II : Systems in triangular quasi-quadratic form and description of residually free groups, J. Algebra 200 (1998) 517 | DOI

[16] M Mihalik, S Tschantz, Strong accessibility of Coxeter groups over minimal splittings, J. Pure Appl. Algebra 216 (2012) 1102 | DOI

[17] D Segal, Polycyclic groups, 82, Cambridge University Press (1983) | DOI

[18] Z Sela, Diophantine geometry over groups, I : Makanin–Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. 93 (2001) 31 | DOI