Twisted Brin–Thompson groups

Belk, James¹ ; Zaremsky, Matthew C B²

¹ School of Mathematics and Statistics, University of St Andrews, St Andrews, United Kingdom
² Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY, United States

Geometry & topology, Tome 26 (2022) no. 3, pp. 1189-1223.

Résumé

We construct a family of infinite simple groups that we call twisted Brin–Thompson groups, generalizing Brin’s higher-dimensional Thompson groups $sV$ for $s\in \mathbb{N}$. We use twisted Brin–Thompson groups to prove a variety of results regarding simple groups. For example, we prove that every finitely generated group embeds quasi-isometrically as a subgroup of a two-generated simple group, strengthening a result of Bridson. We also produce examples of simple groups that contain every $sV$ and hence every right-angled Artin group, including examples of type $F_{\infty }$ and a family of examples of type $F_{n-1}$ but not of type $F_n$ for arbitrary $n\in \mathbb{N}$. This provides the second known infinite family of simple groups distinguished by their finiteness properties.

Détail

DOI : 10.2140/gt.2022.26.1189

Classification : 20F65, 20E32, 57M07
Keywords: Thompson group, finiteness properties, simple group, right-angled Artin group, quasi-isometry, oligomorphic, Cantor space

Bibliographie

[1] J M Alonso, Finiteness conditions on groups and quasi-isometries, J. Pure Appl. Algebra 95 (1994) 121 | DOI

[2] L Bartholdi, Y De Cornulier, D H Kochloukova, Homological finiteness properties of wreath products, Q. J. Math. 66 (2015) 437 | DOI

[3] J Belk, C Bleak, F Matucci, Embedding right-angled Artin groups into Brin–Thompson groups, Math. Proc. Cambridge Philos. Soc. 169 (2020) 225 | DOI

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

[5] J Belk, F Matucci, Röver’s simple group is of type F∞, Publ. Mat. 60 (2016) 501 | DOI

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

[7] A Björner, L Lovász, S T Vrećica, R T Živaljević, Chessboard complexes and matching complexes, J. Lond. Math. Soc. 49 (1994) 25 | DOI

[8] C Bleak, L Elliott, J Hyde, Sufficient conditions for a group of homeomorphisms of the Cantor set to be two-generated, preprint (2020)

[9] C Bleak, D Lanoue, A family of non-isomorphism results, Geom. Dedicata 146 (2010) 21 | DOI

[10] M R Bridson, Controlled embeddings into groups that have no non-trivial finite quotients, from: "The Epstein birthday schrift" (editors I Rivin, C Rourke, C Series), Geom. Topol. Monogr. 1, Geom. Topol. Publ. (1998) 99 | DOI

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

[12] M G Brin, Presentations of higher dimensional Thompson groups, J. Algebra 284 (2005) 520 | DOI

[13] M G Brin, On the baker’s map and the simplicity of the higher dimensional Thompson groups nV , Publ. Mat. 54 (2010) 433 | DOI

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

[15] K S Brown, The geometry of finitely presented infinite simple groups, from: "Algorithms and classification in combinatorial group theory" (editors G Baumslag, C F Miller III), Math. Sci. Res. Inst. Publ. 23, Springer (1992) 121 | DOI

[16] K S Brown, R Geoghegan, An infinite-dimensional torsion-free FP∞ group, Invent. Math. 77 (1984) 367 | DOI

[17] 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

[18] P J Cameron, Oligomorphic permutation groups, 152, Cambridge Univ. Press (1990) | DOI

[19] J W Cannon, W J Floyd, W R Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. 42 (1996) 215

[20] P E Caprace, B Rémy, Simplicity and superrigidity of twin building lattices, Invent. Math. 176 (2009) 169 | DOI

[21] P E Caprace, B Rémy, Non-distortion of twin building lattices, Geom. Dedicata 147 (2010) 397 | DOI

[22] A Darbinyan, M Steenbock, Embeddings into left-orderable simple groups, preprint (2020)

[23] M G Fluch, M Marschler, S Witzel, M C B Zaremsky, The Brin–Thompson groups sV are of type F∞, Pacific J. Math. 266 (2013) 283 | DOI

[24] A P Goryushkin, Imbedding of countable groups in 2–generated simple groups, Mat. Zametki 16 (1974) 231

[25] P Hall, On the embedding of a group in a join of given groups, J. Austral. Math. Soc. 17 (1974) 434 | DOI

[26] J Hennig, F Matucci, Presentations for the higher-dimensional Thompson groups nV , Pacific J. Math. 257 (2012) 53 | DOI

[27] J T Hyde, Constructing 2–generated subgroups of the group of homeomorphisms of Cantor space, PhD thesis, University of St Andrews (2017)

[28] D H Kochloukova, C Martínez-Pérez, B E A Nucinkis, Cohomological finiteness properties of the Brin–Thompson–Higman groups 2V and 3V , Proc. Edinb. Math. Soc. 56 (2013) 777 | DOI

[29] D Quillen, Homotopy properties of the poset of nontrivial p–subgroups of a group, Adv. Math. 28 (1978) 101 | DOI

[30] P E Schupp, Embeddings into simple groups, J. Lond. Math. Soc. 13 (1976) 90 | DOI

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

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

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

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