\(C^1\) actions on the circle of finite index subgroups of \(\mathrm{Mod}(\Sigma_g)\), \(\mathrm{Aut}(F_n)\), and \(\mathrm{Out}(F_n)\). (English) Zbl 1501.37039

Parwani, Kamlesh

Rocky Mt. J. Math. 52, No. 3, 1021-1029 (2022).

Summary: Let \(\Sigma_g\) be a closed, connected, and oriented surface of genus \(g \geq 24\), and let \(\Gamma\) be a finite index subgroup of the mapping class group \(\mathrm{Mod}(\Sigma_g)\) that contains the Torelli group \(\mathscr{I}(\Sigma_g)\). Then any orientation-preserving \(C^1\) action of \(\Gamma\) on the circle cannot be faithful.
We also show that if \(\Gamma\) is a finite index subgroup of \(\mathrm{Aut}(F_n)\), when \(n \geq 8\), that contains the subgroup of IA-automorphisms, then any orientation-preserving \(C^1\) action of \(\Gamma\) on the circle cannot be faithful.
Similarly, if \(\Gamma\) is a finite index subgroup of \(\mathrm{Out}(F_n)\), when \(n \geq 8\), that contains the Torelli group \(\mathscr{T}_n\), then any orientation preserving \(C^1\) action of \(\Gamma\) on the circle cannot be faithful.
In fact, when \(n \geq 10\), any orientation-preserving \(C^1\) action of a finite index subgroup of \(\mathrm{Aut}(F_n)\) or \(\mathrm{Out}(F_n)\) on the circle cannot be faithful.

MSC:

37E10	Dynamical systems involving maps of the circle
20F65	Geometric group theory
57K20	2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)

Keywords:

circle diffeomorphisms; group actions; mapping class group

References:

[1]	H. Baik, S. Kim, and T. Koberda, “Unsmoothable group actions on compact one-manifolds”, J. Eur. Math. Soc. (JEMS) 21:8 (2019), 2333-2353. · Zbl 1454.20074 · doi:10.4171/jems/886
[2]	B. Deroin, V. Kleptsyn, and A. Navas, “Sur la dynamique unidimensionnelle en régularité intermédiaire”, Acta Math. 199:2 (2007), 199-262. · Zbl 1139.37025 · doi:10.1007/s11511-007-0020-1
[3]	M. Ershov and S. He, “On finiteness properties of the Johnson filtrations”, Duke Math. J. 167:9 (2018), 1713-1759. · Zbl 1498.20082 · doi:10.1215/00127094-2018-0005
[4]	B. Farb and D. Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press, Princeton, NJ, 2012. · Zbl 1245.57002
[5]	E. Ghys, “Actions de réseaux sur le cercle”, Invent. Math. 137:1 (1999), 199-231. · Zbl 0995.57006 · doi:10.1007/s002220050329
[6]	E. Ghys, “Groups acting on the circle”, Enseign. Math. (2) 47:3-4 (2001), 329-407. · Zbl 1044.37033
[7]	R. M. Hain, “Torelli groups and geometry of moduli spaces of curves”, pp. 97-143 in Current topics in complex algebraic geometry (Berkeley, CA, 1992/93), edited by H. Clemens and J. Kollár, Math. Sci. Res. Inst. Publ. 28, Cambridge Univ. Press, Cambridge, 1995. · Zbl 0868.14006
[8]	J. L. Harer, “Stability of the homology of the moduli spaces of Riemann surfaces with spin structure”, Math. Ann. 287:2 (1990), 323-334. · Zbl 0715.57004 · doi:10.1007/BF01446896
[9]	N. V. Ivanov, “Mapping class groups”, pp. 523-633 in Handbook of geometric topology, edited by R. J. Daverman and R. B. Sher, North-Holland, Amsterdam, 2002. · Zbl 1002.57001
[10]	D. L. Johnson, “Homeomorphisms of a surface which act trivially on homology”, Proc. Amer. Math. Soc. 75:1 (1979), 119-125. · Zbl 0407.57003 · doi:10.2307/2042686
[11]	M. Kaluba, D. Kielak, and P. W. Nowak, “On property (T) for \[\text{Aut}(F_n)\] and \[\text{SL}_n(\mathbb{Z})\]”, Ann. of Math. (2) 193:2 (2021), 539-562. · Zbl 1483.22006 · doi:10.4007/annals.2021.193.2.3
[12]	S. Kim, T. Koberda, and C. Rivas, “Virtual critical regularity of mapping class group actions on the circle”, preprint, 2021.
[13]	K. Mann and M. Wolff, “Rigidity of mapping class group actions on \[S^1\]”, Geom. Topol. 24:3 (2020), 1211-1223. · Zbl 1454.57014 · doi:10.2140/gt.2020.24.1211
[14]	J. D. McCarthy, “On the first cohomology group of cofinite subgroups in surface mapping class groups”, Topology 40:2 (2001), 401-418. · Zbl 0976.14023 · doi:10.1016/S0040-9383(99)00066-X
[15]	A. Navas, “Actions de groupes de Kazhdan sur le cercle”, Ann. Sci. École Norm. Sup. (4) 35:5 (2002), 749-758. · Zbl 1028.58010 · doi:10.1016/S0012-9593(02)01107-2
[16]	J. Nielsen, “Die Isomorphismengruppe der freien Gruppen”, Math. Ann. 91:3-4 (1924), 169-209. · JFM 50.0078.04 · doi:10.1007/BF01556078
[17]	K. Parwani, “\[{C^1}\] actions on the mapping class groups on the circle”, Algebr. Geom. Topol. 8:2 (2008), 935-944. · Zbl 1155.37028 · doi:10.2140/agt.2008.8.935
[18]	D. Pixton, “Nonsmoothable, unstable group actions”, Trans. Amer. Math. Soc. 229 (1977), 259-268. · Zbl 0361.58005 · doi:10.2307/1998509
[19]	W. P. Thurston, “A generalization of the Reeb stability theorem”, Topology 13 (1974), 347-352. · Zbl 0305.57025 · doi:10.1016/0040-9383(74)90025-1
[20]	K. Vogtmann, “Automorphisms of free groups and outer space”, pp. 1-31 in Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), vol. 94, 2002. · Zbl 1017.20035