Induced quasicocycles on groups with hyperbolically embedded subgroups

Hull, Michael¹ ; Osin, Denis²

¹Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA
²Mathematics Department, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA

Algebraic and Geometric Topology, Tome 13 (2013) no. 5, pp. 2635-2665

Résumé

Let G be a group, H a hyperbolically embedded subgroup of G, V a normed G–module, U an H–invariant submodule of V . We propose a general construction which allows to extend 1–quasicocycles on H with values in U to 1–quasicocycles on G with values in V . As an application, we show that every group G with a nondegenerate hyperbolically embedded subgroup has dimHb2(G,ℓp(G)) = ∞ for p ≥ 1. This covers many previously known results in a uniform way. Applying our extension to quasimorphisms and using Bavard duality, we also show that hyperbolically embedded subgroups are undistorted with respect to the stable commutator length.

Détail

DOI : 10.2140/agt.2013.13.2635

Classification : 20F65, 20F67, 20J06, 43A15, 57M07
Keywords: hyperbolic space, hyperbolically embedded subgroups, left regular representation, quasicocycle, bounded cohomology, stable commutator length

Bibliographie

[1] C Bavard, Longueur stable des commutateurs, Enseign. Math. 37 (1991) 109

[2] M Bestvina, K Bromberg, K Fujiwara, Bounded cohomology via quasi-trees

[3] M Bestvina, M Feighn, A hyperbolic $\mathrm{Out}(F_n)$–complex, Groups Geom. Dyn. 4 (2010) 31

[4] M Bestvina, K Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002) 69

[5] B H Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012) 1250016, 66

[6] R Brooks, Some remarks on bounded cohomology, from: "Riemann surfaces and related topics" (editors I Kra, B Maskit), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 53

[7] D Calegari, scl, MSJ Memoirs 20, Mathematical Society of Japan (2009)

[8] D Calegari, K Fujiwara, Stable commutator length in word-hyperbolic groups, Groups Geom. Dyn. 4 (2010) 59

[9] I Chifan, T Sinclair, On the structural theory of ${\mathrm{II}}_1$ factors of negatively curved groups, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013) 1

[10] F Dahmani, V Guirardel, D Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces

[11] D B A Epstein, K Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997) 1275

[12] L Fuchs, Abelian groups, Publishing House of the Hungarian Academy of Sciences, Budapest (1958) 367

[13] U Hamenstädt, Lines of minima in outer space

[14] U Hamenstädt, Bounded cohomology and isometry groups of hyperbolic spaces, J. Eur. Math. Soc. $($JEMS$)$ 10 (2008) 315

[15] U Hamenstädt, Isometry groups of proper CAT(0)–spaces of rank one, Groups Geom. Dyn. 6 (2012) 579

[16] R C Lyndon, P E Schupp, Combinatorial group theory, Ergeb. Math. Grenzgeb. 89, Springer (1977)

[17] I Mineyev, N Monod, Y Shalom, Ideal bicombings for hyperbolic groups and applications, Topology 43 (2004) 1319

[18] N Monod, Continuous bounded cohomology of locally compact groups, Lecture Notes in Mathematics 1758, Springer (2001)

[19] N Monod, An invitation to bounded cohomology, from: "Proc. ICM, Vol. II" (editors M Sanz-Solé, J Soria, J L Varona, J Verdera), Eur. Math. Soc., Zürich (2006) 1183

[20] N Monod, Y Shalom, Negative curvature from a cohomological viewpoint and cocycle superrigidity, C. R. Math. Acad. Sci. Paris 337 (2003) 635

[21] N Monod, Y Shalom, Orbit equivalence rigidity and bounded cohomology, Ann. of Math. 164 (2006) 825

[22] D V Osin, Elementary subgroups of relatively hyperbolic groups and bounded generation, Internat. J. Algebra Comput. 16 (2006) 99

[23] D Osin, Small cancellations over relatively hyperbolic groups and embedding theorems, Ann. of Math. 172 (2010) 1

[24] S Popa, Deformation and rigidity for group actions and von Neumann algebras, from: "Proc ICM, Vol. I" (editors M Sanz-Solé, J Soria, J L Varona, J Verdera), Eur. Math. Soc., Zürich (2007) 445

[25] A Sisto, Contracting elements and random walks

[26] A Thom, Low degree bounded cohomology and $L^2$–invariants for negatively curved groups, Groups Geom. Dyn. 3 (2009) 343