Virtually splitting the map from 𝐴𝑢𝑡(𝐺) to 𝑂𝑢𝑡(𝐺)

Carette, Mathieu

doi:10.1090/S0002-9939-2014-12278-7

Virtually splitting the map from $\operatorname {Aut}(G)$ to $\operatorname {Out}(G)$
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by Mathieu Carette

Proc. Amer. Math. Soc. 143 (2015), 543-554

DOI: https://doi.org/10.1090/S0002-9939-2014-12278-7

Published electronically: October 10, 2014

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Abstract:

We give an elementary criterion on a group $G$ for the map $\operatorname {Aut}(G)$ $\to \operatorname {Out}(G)$ to split virtually. This criterion applies to many residually finite $\operatorname {CAT}(0)$ groups and hyperbolic groups, and in particular to all finitely generated Coxeter groups. As a consequence the outer automorphism group of any finitely generated Coxeter group is residually finite and virtually torsion-free.

References

Gilbert Baumslag, Automorphism groups of residually finite groups, J. London Math. Soc. 38 (1963), 117–118. MR 146271, DOI 10.1112/jlms/s1-38.1.117
Mladen Bestvina, Questions in geometric group theory, Available at http://www.math.utah.edu/˜bestvina.
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Brian H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998), no. 2, 145–186. MR 1638764, DOI 10.1007/BF02392898
Inna Bumagin and Daniel T. Wise, Every group is an outer automorphism group of a finitely generated group, J. Pure Appl. Algebra 200 (2005), no. 1-2, 137–147. MR 2142354, DOI 10.1016/j.jpaa.2004.12.033
Mathieu Carette, The automorphism group of accessible groups, J. Lond. Math. Soc. (2) 84 (2011), no. 3, 731–748. MR 2855799, DOI 10.1112/jlms/jdr029
Pierre-Emmanuel Caprace and Ashot Minasyan, On conjugacy separability of some Coxeter groups and parabolic-preserving automorphisms, to appear in Illinois J. Math., http://arxiv.org/abs/1210.4328, 2012.
Pierre-Emmanuel Caprace and Piotr Przytycki, Twist-rigid Coxeter groups, Geom. Topol. 14 (2010), no. 4, 2243–2275. MR 2740646, DOI 10.2140/gt.2010.14.2243
Ruth Charney and Karen Vogtmann, Subgroups and quotients of automorphism groups of RAAGs, Low-dimensional and symplectic topology, Proc. Sympos. Pure Math., vol. 82, Amer. Math. Soc., Providence, RI, 2011, pp. 9–27. MR 2768650, DOI 10.1090/pspum/082/2768650
Sheila C. Chagas and Pavel A. Zalesskii, Finite index subgroups of conjugacy separable groups, Forum Math. 21 (2009), no. 2, 347–353. MR 2503309, DOI 10.1515/FORUM.2009.016
Michael W. Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, NJ, 2008. MR 2360474
Vincent Guirardel and Gilbert Levitt, Splittings and automorphisms of relatively hyperbolic groups, to appear in Groups, Geom. and Dyn., http://arxiv.org/abs/1212.1434v1, 2012.
A. V. Goryaga, Example of a finite extension of an FAC-group that is not an FAC-group, Sibirsk. Mat. Zh. 27 (1986), no. 3, 203–205, 225 (Russian). MR 853899
Mauricio Gutierrez, Adam Piggott, and Kim Ruane, On the automorphisms of a graph product of abelian groups, Groups Geom. Dyn. 6 (2012), no. 1, 125–153. MR 2888948, DOI 10.4171/GGD/153
Edna K. Grossman, On the residual finiteness of certain mapping class groups, J. London Math. Soc. (2) 9 (1974/75), 160–164. MR 405423, DOI 10.1112/jlms/s2-9.1.160
Lynne D. James, Complexes and Coxeter groups—operations and outer automorphisms, J. Algebra 113 (1988), no. 2, 339–345. MR 929764, DOI 10.1016/0021-8693(88)90163-9
Gilbert Levitt, Automorphisms of hyperbolic groups and graphs of groups, Geom. Dedicata 114 (2005), 49–70. MR 2174093, DOI 10.1007/s10711-004-1492-1
Gilbert Levitt and Ashot Minasyan, Residual properties of automorphism groups of (relatively) hyperbolic groups, to appear in Geom. and Topol., http://arxiv.org/abs/1305.5403, 2013.
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
Alexander Lubotzky, Normal automorphisms of free groups, J. Algebra 63 (1980), no. 2, 494–498. MR 570726, DOI 10.1016/0021-8693(80)90086-1
G. Mess, Unit tangent bundle subgroups of mapping class groups, IHES preprint, 1990.
Ashot Minasyan, Groups with finitely many conjugacy classes and their automorphisms, Comment. Math. Helv. 84 (2009), no. 2, 259–296. MR 2495795, DOI 10.4171/CMH/162
Ashot Minasyan, Hereditary conjugacy separability of right-angled Artin groups and its applications, Groups Geom. Dyn. 6 (2012), no. 2, 335–388. MR 2914863, DOI 10.4171/GGD/160
Armando Martino and Ashot Minasyan, Conjugacy in normal subgroups of hyperbolic groups, Forum Math. 24 (2012), no. 5, 889–910. MR 2988566, DOI 10.1515/form.2011.089
A. Minasyan and D. Osin, Normal automorphisms of relatively hyperbolic groups, Trans. Amer. Math. Soc. 362 (2010), no. 11, 6079–6103. MR 2661509, DOI 10.1090/S0002-9947-2010-05067-6
Nicolas Monod, Superrigidity for irreducible lattices and geometric splitting, J. Amer. Math. Soc. 19 (2006), no. 4, 781–814. MR 2219304, DOI 10.1090/S0894-0347-06-00525-X
Gabor Moussong, Hyperbolic Coxeter groups, ProQuest LLC, Ann Arbor, MI, 1988. Thesis (Ph.D.)–The Ohio State University. MR 2636665
Bernhard Mühlherr, Automorphisms of graph-universal Coxeter groups, J. Algebra 200 (1998), no. 2, 629–649. MR 1610676, DOI 10.1006/jabr.1997.7230
Bernhard Mühlherr, The isomorphism problem for Coxeter groups, The Coxeter legacy, Amer. Math. Soc., Providence, RI, 2006, pp. 1–15. MR 2209018
B. B. Newman, Some results on one-relator groups, Bull. Amer. Math. Soc. 74 (1968), 568–571. MR 222152, DOI 10.1090/S0002-9904-1968-12012-9
Frédéric Paulin, Outer automorphisms of hyperbolic groups and small actions on $\textbf {R}$-trees, Arboreal group theory (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 331–343. MR 1105339, DOI 10.1007/978-1-4612-3142-4_{1}2
Kim E. Ruane, Dynamics of the action of a $\textrm {CAT}(0)$ group on the boundary, Geom. Dedicata 84 (2001), no. 1-3, 81–99. MR 1825346, DOI 10.1023/A:1010301824765
Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank $1$ Lie groups. II, Geom. Funct. Anal. 7 (1997), no. 3, 561–593. MR 1466338, DOI 10.1007/s000390050019
Jacques Tits, Sur le groupe des automorphismes de certains groupes de Coxeter, J. Algebra 113 (1988), no. 2, 346–357 (French). MR 929765, DOI 10.1016/0021-8693(88)90164-0
Daniel T. Wise, Research announcement: the structure of groups with a quasiconvex hierarchy, Electron. Res. Announc. Math. Sci. 16 (2009), 44–55. MR 2558631, DOI 10.3934/era.2009.16.44