Asymptotically rigid mapping class groups, I : Finiteness properties of braided Thompson’s and Houghton’s groups

Genevois, Anthony¹ ; Lonjou, Anne² ; Urech, Christian³

¹ Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, Montpellier, France
² Département de Mathématiques, Faculté des Sciences d’Orsay, Université Paris-Saclay, Orsay, France
³ Institut de Mathématiques, École Polytechniqe Fédérale de Lausanne, Lausanne, Switzerland

Geometry & topology, Tome 26 (2022) no. 3, pp. 1385-1434.

Résumé

We study the asymptotically rigid mapping class groups of infinitely punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy–Thompson groups $T^{♯},T^{\ast }$ introduced by Funar and Kapoudjian, and the braided Houghton groups $\mathrm{br}{H}_n$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube stabilizers isomorphic to finite extensions of braid groups. As an application, we prove conjectures of Funar–Kapoudjian and Degenhardt by showing that $T^{♯}$ and $T^{\ast }$ are of type $F_{\infty }$ and that $\mathrm{br}{H}_n$ is of type $F_{n-1}$ but not of type $F_n$.

Détail

DOI : 10.2140/gt.2022.26.1385

Keywords: Thompson groups, Houghton groups, braid groups, big mapping class groups, asymptotically rigid mapping class groups, cube complexes

Bibliographie

[1] J Aramayona, L Funar, Asymptotic mapping class groups of closed surfaces punctured along Cantor sets, Mosc. Math. J. 21 (2021) 1 | DOI

[2] J Belk, B Forrest, Rearrangement groups of fractals, Trans. Amer. Math. Soc. 372 (2019) 4509 | DOI

[3] J Belk, M C B Zaremsky, Twisted Brin–Thompson groups, Geom. Topol. 26 (2022) 1189 | DOI

[4] R W Bell, D Margalit, Braid groups and the co-Hopfian property, J. Algebra 303 (2006) 275 | DOI

[5] M Bestvina, N Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997) 445 | DOI

[6] T Brady, J Burillo, S Cleary, M Stein, Pure braid subgroups of braided Thompson’s groups, Publ. Mat. 52 (2008) 57 | DOI

[7] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999) | DOI

[8] M G Brin, Higher dimensional Thompson groups, Geom. Dedicata 108 (2004) 163 | DOI

[9] M G Brin, The algebra of strand splitting, I : A braided version of Thompson’s group V , J. Group Theory 10 (2007) 757 | DOI

[10] K S Brown, Cohomology of groups, 87, Springer (1982) | DOI

[11] K S Brown, Finiteness properties of groups, J. Pure Appl. Algebra 44 (1987) 45 | DOI

[12] K U Bux, Tangling and braiding the chessboard complex, preprint (2003)

[13] K U Bux, Higher finiteness properties of braided groups, from: "Spectral structures and topological methods in mathematics" (editors M Baake, F Götze, W Hoffmann), Eur. Math. Soc. (2019) 299 | DOI

[14] K U Bux, Arc matching complexes and finiteness properties, Oberwolfach Rep. 17 (2021) 1121

[15] K U Bux, M G Fluch, M Marschler, S Witzel, M C B Zaremsky, The braided Thompson’s groups are of type F∞, J. Reine Angew. Math. 718 (2016) 59 | DOI

[16] F Degenhardt, Endlichkeitseigenschaften gewisser Gruppen von Zöpfen unendlicher Ordnung, PhD thesis, Goethe-Universität Frankfurt am Main (2000)

[17] P Dehornoy, The group of parenthesized braids, Adv. Math. 205 (2006) 354 | DOI

[18] C Druţu, M Kapovich, Geometric group theory, 63, Amer. Math. Soc. (2018) | DOI

[19] S Eilenberg, Sur les transformations périodiques de la surface de sphère, Fund. Math. 22 (1934) 28 | DOI

[20] D S Farley, Homological and finiteness properties of picture groups, Trans. Amer. Math. Soc. 357 (2005) 3567 | DOI

[21] L Funar, Braided Houghton groups as mapping class groups, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. 53 (2007) 229

[22] L Funar, C Kapoudjian, On a universal mapping class group of genus zero, Geom. Funct. Anal. 14 (2004) 965 | DOI

[23] L Funar, C Kapoudjian, The braided Ptolemy–Thompson group is finitely presented, Geom. Topol. 12 (2008) 475 | DOI

[24] L Funar, C Kapoudjian, The braided Ptolemy–Thompson group is asynchronously combable, Comment. Math. Helv. 86 (2011) 707 | DOI

[25] L Funar, C Kapoudjian, V Sergiescu, Asymptotically rigid mapping class groups and Thompson’s groups, from: "Handbook of Teichmüller theory, III" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 17, Eur. Math. Soc. (2012) 595 | DOI

[26] A Genevois, A Lonjou, C Urech, Asymptotically rigid mapping class groups, II: Strand diagrams and nonpositive curvature, preprint (2021)

[27] A Genevois, A Lonjou, C Urech, Asymptotically rigid mapping class groups, III: Explicit presentations, in preparation

[28] V Guba, M Sapir, Diagram groups, 620, Amer. Math. Soc. (1997) | DOI

[29] J L Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. 121 (1985) 215 | DOI

[30] A Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991) 189 | DOI

[31] G Higman, Finitely presented infinite simple groups, 8, Aust. Nat. Univ. (1974)

[32] C H Houghton, The first cohomology of a group with permutation module coefficients, Arch. Math. (Basel) 31 (1978) 254 | DOI

[33] B Hughes, Local similarities and the Haagerup property, Groups Geom. Dyn. 3 (2009) 299 | DOI

[34] Y Lodha, J T Moore, A nonamenable finitely presented group of piecewise projective homeomorphisms, Groups Geom. Dyn. 10 (2016) 177 | DOI

[35] N Monod, Groups of piecewise projective homeomorphisms, Proc. Natl. Acad. Sci. USA 110 (2013) 4524 | DOI

[36] V V Nekrashevych, Cuntz–Pimsner algebras of group actions, J. Operator Theory 52 (2004) 223

[37] B E A Nucinkis, S St. John-Green, Quasi-automorphisms of the infinite rooted 2–edge-coloured binary tree, Groups Geom. Dyn. 12 (2018) 529 | DOI

[38] C E Röver, Constructing finitely presented simple groups that contain Grigorchuk groups, J. Algebra 220 (1999) 284 | DOI

[39] R Skipper, S Witzel, M C B Zaremsky, Simple groups separated by finiteness properties, Invent. Math. 215 (2019) 713 | DOI

[40] M Stein, Groups of piecewise linear homeomorphisms, Trans. Amer. Math. Soc. 332 (1992) 477 | DOI

[41] W Thumann, Operad groups and their finiteness properties, Adv. Math. 307 (2017) 417 | DOI

[42] S Witzel, Classifying spaces from Ore categories with Garside families, Algebr. Geom. Topol. 19 (2019) 1477 | DOI

[43] S Witzel, M C B Zaremsky, Thompson groups for systems of groups, and their finiteness properties, Groups Geom. Dyn. 12 (2018) 289 | DOI