Symmetric automorphisms of free groups, BNSR-invariants, and finiteness properties. (English) Zbl 06965593

Summary: The BNSR-invariants of a group \(G\) are a sequence \(\Sigma^{1}(G)\supseteq \Sigma^{2}(G)\supseteq \cdots \) of geometric invariants that reveal important information about finiteness properties of certain subgroups of \(G\). We consider the symmetric automorphism group \(\operatorname{\Sigma Aut}_{n}\) and pure symmetric automorphism group \(\operatorname{P\!\operatorname{\Sigma Aut}}_{n}\) of the free group \(F_{n}\) and inspect their BNSR-invariants. We prove that for \(n\geq 2\), all the “positive” and “negative” character classes of \(\operatorname{P\!\operatorname{\Sigma Aut}}_{n}\) lie in \(\Sigma^{n-2}(\operatorname{P\!\operatorname{\Sigma Aut}}_{n})\setminus \Sigma^{n-1}(\operatorname{P\!\operatorname{\Sigma Aut}}_{n})\). We use this to prove that for \(n\geq 2\), \(\Sigma^{n-2}(\operatorname{\Sigma Aut}_{n})\) equals the full character sphere \(S^{0}\) of \(\operatorname{\Sigma Aut}_{n}\) but \(\Sigma^{n-1}(\operatorname{\Sigma Aut}_{n})\) is empty, so in particular the commutator subgroup \(\operatorname{\Sigma Aut}_{n}'\) is of type \(\operatorname{F}_{n-2}\) but not \(\operatorname{F}_{n-1}\). Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.


20-XX Group theory and generalizations
20F65 Geometric group theory
20F28 Automorphism groups of groups
57M07 Topological methods in group theory
Full Text: DOI arXiv Euclid


[1] [BB97] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445–470. · Zbl 0888.20021
[2] [BBM07] M. Bestvina, K.-U. Bux, and D. Margalit, Dimension of the Torelli group for \(\text{Out}(F_{n})\), Invent. Math. 170 (2007), no. 1, 1–32. · Zbl 1135.20026
[3] [BGK10] R. Bieri, R. Geoghegan, and D. H. Kochloukova, The Sigma invariants of Thompson’s group \(F\), Groups Geom. Dyn. 4 (2010), no. 2, 263–273. · Zbl 1214.20048
[4] [BNS87] R. Bieri, W. D. Neumann, and R. Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987), no. 3, 451–477. · Zbl 0642.57002
[5] [BR88] R. Bieri and B. Renz, Valuations on free resolutions and higher geometric invariants of groups, Comment. Math. Helv. 63 (1988), no. 3, 464–497. · Zbl 0654.20029
[6] [BLVŽ94] A. Björner, L. Lovász, S. T. Vrećica, and R. T. Živaljević, Chessboard complexes and matching complexes, J. Lond. Math. Soc. (2) 49 (1994), no. 1, 25–39. · Zbl 0790.57014
[7] [BMMM01] N. Brady, J. McCammond, J. Meier, and A. Miller, The pure symmetric automorphisms of a free group form a duality group, J. Algebra 246 (2001), no. 2, 881–896. · Zbl 0995.20010
[8] [Bro87] K. S. Brown, Trees, valuations, and the Bieri–Neumann–Strebel invariant, Invent. Math. 90 (1987), no. 3, 479–504. · Zbl 0663.20033
[9] [Bux04] K.-U. Bux, Finiteness properties of soluble arithmetic groups over global function fields, Geom. Topol. 8 (2004), 611–644 (electronic). · Zbl 1066.20049
[10] [BG99] K.-U. Bux and C. Gonzalez, The Bestvina–Brady construction revisited: geometric computation of \(Σ \)-invariants for right-angled Artin groups, J. Lond. Math. Soc. (2) 60 (1999), no. 3, 793–801. · Zbl 1025.20026
[11] [Col89] D. J. Collins, Cohomological dimension and symmetric automorphisms of a free group, Comment. Math. Helv. 64 (1989), no. 1, 44–61. · Zbl 0669.20027
[12] [CV86] M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91–119. · Zbl 0589.20022
[13] [Dam16] C. Damiani, A journey through loop braid groups, 2016, arXiv:1605.02323. · Zbl 1403.20052
[14] [Gri13] J. T. Griffin, Diagonal complexes and the integral homology of the automorphism group of a free product, Proc. Lond. Math. Soc. (3) 106 (2013), no. 5, 1087–1120. · Zbl 1278.20070
[15] [HV98] A. Hatcher and K. Vogtmann, Cerf theory for graphs, J. Lond. Math. Soc. (2) 58 (1998), no. 3, 633–655. · Zbl 0922.57001
[16] [JMM06] C. Jensen, J. McCammond, and J. Meier, The integral cohomology of the group of loops, Geom. Topol. 10 (2006), 759–784. · Zbl 1162.20036
[17] [Jen02] C. A. Jensen, Contractibility of fixed point sets of auter space, Topology Appl. 119 (2002), no. 3, 287–304. · Zbl 0997.20042
[18] [KMM15] N. Koban, J. McCammond, and J. Meier, The BNS-invariant for the pure braid groups, Groups Geom. Dyn. 9 (2015), no. 3, 665–682. · Zbl 1326.20045
[19] [KP14] N. Koban and A. Piggott, The Bieri–Neumann–Strebel invariant of the pure symmetric automorphisms of a right-angled Artin group, Illinois J. Math. 58 (2014), no. 1, 27–41. · Zbl 1332.20044
[20] [Koc12] D. H. Kochloukova, On the \(Σ^{2}\)-invariants of the generalised R. Thompson groups of type \(F\), J. Algebra 371 (2012), 430–456. · Zbl 1279.20068
[21] [McC86] J. McCool, On basis-conjugating automorphisms of free groups, Canad. J. Math. 38 (1986), no. 6, 1525–1529. · Zbl 0613.20024
[22] [MMV98] J. Meier, H. Meinert, and L. VanWyk, Higher generation subgroup sets and the \(Σ \)-invariants of graph groups, Comment. Math. Helv. 73 (1998), no. 1, 22–44. · Zbl 0899.57001
[23] [OK00] L. A. Orlandi-Korner, The Bieri–Neumann–Strebel invariant for basis-conjugating automorphisms of free groups, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1257–1262. · Zbl 0963.20016
[24] [Pet10] A. Pettet, Finiteness properties for a subgroup of the pure symmetric automorphism group, C. R. Math. Acad. Sci. Paris 348 (2010), no. 3–4, 127–130. · Zbl 1220.20028
[25] [Qui78] D. Quillen, Homotopy properties of the poset of nontrivial \(p\)-subgroups of a group, Adv. Math. 28 (1978), no. 2, 101–128. · Zbl 0388.55007
[26] [Sav96] A. G. Savushkina, On a group of conjugating automorphisms of a free group, Mat. Zametki 60 (1996), no. 1, 92–108, 159.
[27] [WZ16] S. Witzel and M. C. B. Zaremsky, The Basilica Thompson group is not finitely presented, submitted, 2016, arXiv:1603.01150.
[28] [Zar17] M. C. B. Zaremsky, On the \(Σ \)-invariants of generalized Thompson groups and Houghton groups, Int. Math. Res. Not. IMRN 2017 (2017), no. 19, 5861–5896.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.