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Outer space for RAAGs. (English) Zbl 1521.20072

Summary: For any right-angled Artin group \({A_{\Gamma}}\), we construct a finite-dimensional space \({\mathcal{O}_{\Gamma}}\) on which the group \(\operatorname{Out}({A_{\Gamma}})\) of outer automorphisms of \({A_{\Gamma}}\) acts with finite point stabilizers. We prove that \({\mathcal{O}_{\Gamma}}\) is contractible, so that the quotient is a rational classifying space for \(\operatorname{Out}({A_{\Gamma}})\). The space \({\mathcal{O}_{\Gamma}}\) blends features of the symmetric space of lattices in \({\mathbb{R}^n}\) with those of outer space for the free group \({F_n}\). Points in \({\mathcal{O}_{\Gamma}}\) are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cube complexes, marked by an isomorphism of their fundamental group with \({A_{\Gamma}}\).

MSC:

20F36 Braid groups; Artin groups
20F28 Automorphism groups of groups
20F65 Geometric group theory
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[1] I. AGOL, The virtual Haken conjecture, with appendix “Filling virtually special subgroups” by I. Agol, D. Groves, and J. Manning, Doc. Math. 18 (2013), 1045-1087. · Zbl 1286.57019 · doi:Incorrect DOI link
[2] N. BERGERON and D. T. WISE, A boundary criterion for cubulation, Amer. J. Math. 134 (2012), no. 3, 843-859. · Zbl 1279.20051 · doi:10.1353/ajm.2012.0020
[3] M. BESTVINA and N. BRADY, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445-470. · Zbl 0888.20021 · doi:10.1007/s002220050168
[4] M. BESTVINA and M. FEIGHN, The topology at infinity of \[ \text{Out}({F_n})\], Invent. Math. 140 (2000), no. 3, 651-692. · Zbl 0954.55011 · doi:10.1007/s002220000068
[5] J. BEYRER and E. FIORAVANTI, Cross ratios on \[ \text{CAT}(0)\]cube complexes and marked length-spectrum rigidity, J. Lond. Math. Soc. (2) 104 (2021), no. 5, 1973-2015. · Zbl 07652696 · doi:10.1112/jlms.12489
[6] A. BOREL and J.-P. SERRE, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436-491. · Zbl 0274.22011 · doi:10.1007/BF02566134
[7] C. BREGMAN, Isometry groups of skewed Γ-complexes, preprint, arXiv:2202.09860v1 [math.GT].
[8] M. R. BRIDSON and A. HAEFLIGER, Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wiss. 319, Springer, Berlin, 1999. · Zbl 0988.53001 · doi:10.1007/978-3-662-12494-9
[9] M. R. BRIDSON and K. VOGTMANN, On the geometry of the automorphism group of a free group, Bull. London Math. Soc. 27 (1995), no. 6, 544-552. · Zbl 0836.20045 · doi:10.1112/blms/27.6.544
[10] B. BRÜCK and R. D. WADE, A note on virtual duality and automorphism groups of right-angled Artin groups, preprint, arXiv:2101.08225v1 [math.GR].
[11] R. CHARNEY, J. CRISP, and K. VOGTMANN, Automorphisms of 2-dimensional right-angled Artin groups, Geom. Topol. 11 (2007), 2227-2264. · Zbl 1152.20032 · doi:10.2140/gt.2007.11.2227
[12] R. CHARNEY, N. STAMBAUGH, and K. VOGTMANN, Outer space for untwisted automorphisms of right-angled Artin groups, Geom. Topol. 21 (2017), no. 2, 1131-1178. · Zbl 1405.20028 · doi:10.2140/gt.2017.21.1131
[13] R. CHARNEY and K. VOGTMANN, Finiteness properties of automorphism groups of right-angled Artin groups, Bull. Lond. Math. Soc. 41 (2009), no. 1, 94-102. · Zbl 1244.20036 · doi:10.1112/blms/bdn108
[14] R. CHARNEY and K. VOGTMANN, “Subgroups and quotients of automorphism groups of RAAGs” in Low-Dimensional and Symplectic Topology, Proc. Sympos. Pure Math. 82, Amer. Math. Soc., Providence, 2011, 9-27. · Zbl 1235.20034 · doi:10.1090/pspum/082/2768650
[15] M. T. CLAY, C. J. LEININGER, and J. MANGAHAS, The geometry of right-angled Artin subgroups of mapping class groups, Groups Geom. Dyn. 6 (2012), no. 2, 249-278. · Zbl 1245.57004 · doi:10.4171/GGD/157
[16] J. CRISP and L. PARIS, The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group, Invent. Math. 145 (2001), no. 1, 19-36. · Zbl 1002.20021 · doi:10.1007/s002220100138
[17] C. B. CROKE and B. KLEINER, Spaces with nonpositive curvature and their ideal boundaries, Topology 39 (2000), no. 3, 549-556. · Zbl 0959.53014 · doi:10.1016/S0040-9383(99)00016-6
[18] M. CULLER and K. VOGTMANN, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91-119. · Zbl 0589.20022 · doi:10.1007/BF01388734
[19] M. W. DAVIS and T. JANUSZKIEWICZ, Right-angled Artin groups are commensurable with right-angled Coxeter groups, J. Pure Appl. Algebra 153 (2000), no. 3, 229-235. · Zbl 0982.20022 · doi:10.1016/S0022-4049(99)00175-9
[20] M. B. DAY, Peak reduction and finite presentations for automorphism groups of right-angled Artin groups, Geom. Topol. 13 (2009), no. 2, 817-855. · Zbl 1226.20024 · doi:10.2140/gt.2009.13.817
[21] M. B. DAY, A. W. SALE, and R. D. WADE, Calculating the virtual cohomological dimension of the automorphism group of a RAAG, Bull. Lond. Math. Soc. 53 (2021), no. 1, 259-273. · Zbl 07367034 · doi:10.1112/blms.12418
[22] M. B. DAY and R. D. WADE, Relative automorphism groups of right-angled Artin groups, J. Topol. 12 (2019), no. 3, 759-798. · Zbl 1481.20104 · doi:10.1112/topo.12101
[23] D. B. A. EPSTEIN, J. W. CANNON, D. F. HOLT, S. V. F. LEVY, M. S. PATERSON, and W. P. THURSTON, Word Processing in Groups, Jones and Bartlett, Boston, 1992. · Zbl 0764.20017
[24] F. Haglund and D. T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1551-1620. · Zbl 1155.53025 · doi:10.1007/s00039-007-0629-4
[25] A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002. · Zbl 1044.55001
[26] T. KOBERDA, Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups, Geom. Funct. Anal. 22 (2012), no. 6, 1541-1590. · Zbl 1282.37024 · doi:10.1007/s00039-012-0198-z
[27] M. R. LAURENCE, A generating set for the automorphism group of a graph group, J. Lond. Math. Soc. (2) 52 (1995), no. 2, 318-334. · Zbl 0836.20036 · doi:10.1112/jlms/52.2.318
[28] B. MILLARD and K. VOGTMANN, Cube complexes and abelian subgroups of automorphism groups of RAAGs, Math. Proc. Cambridge Philos. Soc. 170 (2021), no. 3, 1-25. · Zbl 07395446 · doi:10.1017/S0305004119000501
[29] L. MOSHER, Mapping class groups are automatic, Ann. of Math. (2) 142 (1995), no. 2, 303-384. · Zbl 0867.57004 · doi:10.2307/2118637
[30] M. Sageev, “\[ \text{CAT}(\text{0})\] cube complexes and groups” in Geometric Group Theory, IAS/Park City Math. Ser. 21, Amer. Math. Soc., Providence, 2014, 7-54. · Zbl 1440.20015 · doi:10.1090/pcms/021/02
[31] H. SERVATIUS, Automorphisms of graph groups, J. Algebra 126 (1989), no. 1, 34-60. · Zbl 0682.20022 · doi:10.1016/0021-8693(89)90319-0
[32] H. SERVATIUS, C. DROMS, and B. SERVATIUS, Surface subgroups of graph groups, Proc. Amer. Math. Soc. 106 (1989), no. 3, 573-578. · Zbl 0677.20023 · doi:10.2307/2047406
[33] A. VIJAYAN, Compactifying the space of length functions of a right-angled Artin group, Ph.D. dissertation, Brandeis University, Waltham, 2013
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