Limits of (certain) CAT(0) groups, I: Compactification

Groves, Daniel¹

¹Department of Mathematics, California Institute of Technology, Pasadena CA 91125, USA

Algebraic and Geometric Topology, Tome 5 (2005) no. 4, pp. 1325-1364

Résumé

The purpose of this paper is to investigate torsion-free groups which act properly and cocompactly on CAT(0) metric spaces which have isolated flats, as defined by Hruska. Our approach is to seek results analogous to those of Sela, Kharlampovich and Miasnikov for free groups and to those of Sela (and Rips and Sela) for torsion-free hyperbolic groups.

This paper is the first in a series. In this paper we extract an ℝ–tree from an asymptotic cone of certain CAT(0) spaces. This is analogous to a construction of Paulin, and allows a great deal of algebraic information to be inferred, most of which is left to future work.

Détail

DOI : 10.2140/agt.2005.5.1325

Keywords: CAT(0) spaces, isolated flats, limit groups, $\mathbb{R}$–trees

Bibliographie

[1] M Bestvina, Degenerations of the hyperbolic space, Duke Math. J. 56 (1988) 143

[2] M Bestvina, $\mathbb R$–trees in topology, geometry, and group theory, from: "Handbook of geometric topology", North-Holland (2002) 55

[3] M Bestvina, Questions in geometric group theory (2004)

[4] M Bestvina, M Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995) 287

[5] M Bestvina, M Feighn, Notes on Sela's work: Limit groups and Makanin–Razborov diagrams, preprint

[6] L Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume, Math. Ann. 70 (1911) 297

[7] L Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich, Math. Ann. 72 (1912) 400

[8] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer (1999)

[9] M R Bridson, G A Swarup, On Hausdorff–Gromov convergence and a theorem of Paulin, Enseign. Math. $(2)$ 40 (1994) 267

[10] C Champetier, L'espace des groupes de type fini, Topology 39 (2000) 657

[11] C Champetier, V Guirardel, Limit groups as limits of free groups, Israel J. Math. 146 (2005) 1

[12] I Chiswell, Introduction to $\Lambda$–trees, World Scientific Publishing Co. (2001)

[13] L Van Den Dries, A J Wilkie, Gromov's theorem on groups of polynomial growth and elementary logic, J. Algebra 89 (1984) 349

[14] C Druţu, M Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005) 959

[15] 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 Publishers (1992)

[16] R I Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984) 939

[17] D Groves, Limit groups for (certain) CAT(0) groups II: The Hopf property and the shortening argument

[18] G C Hruska, Nonpositively curved spaces with isolated flats, PhD thesis, Cornell University (2002)

[19] G C Hruska, B Kleiner, Hadamard spaces with isolated flats, Geom. Topol. 9 (2005) 1501

[20] M Kapovich, B Leeb, On asymptotic cones and quasi-isometry classes of fundamental groups of 3–manifolds, Geom. Funct. Anal. 5 (1995) 582

[21] O Kharlampovich, A Myasnikov, Elementary theory of free non-abelian groups, J. Algebra 302 (2006) 451

[22] J W Morgan, P B Shalen, Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. $(2)$ 120 (1984) 401

[23] F Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels, Invent. Math. 94 (1988) 53

[24] F Paulin, Outer automorphisms of hyperbolic groups and small actions on $\mathbb{R}$–trees, from: "Arboreal group theory (Berkeley, CA, 1988)", Math. Sci. Res. Inst. Publ. 19, Springer (1991) 331

[25] F Paulin, Sur la théorie élémentaire des groupes libres (d'après Sela), Astérisque (2004) 363

[26] E Rips, Z Sela, Structure and rigidity in hyperbolic groups I, Geom. Funct. Anal. 4 (1994) 337

[27] E Rips, Z Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. $(2)$ 146 (1997) 53

[28] Z Sela, The isomorphism problem for hyperbolic groups I, Ann. of Math. $(2)$ 141 (1995) 217

[29] Z Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups II, Geom. Funct. Anal. 7 (1997) 561

[30] Z Sela, Acylindrical accessibility for groups, Invent. Math. 129 (1997) 527

[31] Z Sela, Endomorphisms of hyperbolic groups I: The Hopf property, Topology 38 (1999) 301

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

[33] Z Sela, Diophantine geometry over groups VI: The elementary theory of a free group, preprint

[34] Z Sela, Diophantine geometry over groups VII: The elementary theory of a hyperbolic group, preprint

[35] Z Sela, Diophantine geometry over groups: a list of research problems

[36] P B Shalen, Dendrology of groups: an introduction, from: "Essays in group theory", Math. Sci. Res. Inst. Publ. 8, Springer (1987) 265

[37] P B Shalen, Dendrology and its applications, from: "Group theory from a geometrical viewpoint (Trieste, 1990)", World Sci. Publ., River Edge, NJ (1991) 543

[38] W P Thurston, Three-dimensional geometry and topology Vol. 1, Princeton Mathematical Series 35, Princeton University Press (1997)

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

[40] D T Wise, Subgroup separability of the figure 8 knot group, Topology 45 (2006) 421