Compact presentability of tree almost automorphism groups
Annales de l'Institut Fourier, Volume 67 (2017) no. 1, p. 329-365
We establish compact presentability, i.e. the locally compact version of finite presentability, for an infinite family of tree almost automorphism groups. Examples covered by our results include Neretin’s group of spheromorphisms, as well as the topologically simple group containing the profinite completion of the Grigorchuk group constructed by Barnea, Ershov and Weigel.We additionally obtain an upper bound on the Dehn function of these groups in terms of the Dehn function of an embedded Higman–Thompson group. This, combined with a result of Guba, implies that the Dehn function of the Neretin group of the regular trivalent tree is polynomially bounded.
Nous prouvons que certains groupes de presqu’automorphismes d’arbres sont compactement présentés. Parmi ces groupes figurent le groupe de Neretin des sphéromorphismes d’un arbre régulier, ainsi que le groupe topologiquement simple contenant le complété profini du groupe de Grigorchuk construit par Barnea, Ershov et Weigel.Nous montrons de plus que la fonction de Dehn de ces groupes est asymptotiquement bornée par la fonction de Dehn du groupe de Higman–Thompson. Combiné à un résultat de Guba, cela implique que la fonction de Dehn du groupe de Neretin de l’arbre trivalent est polynomialement bornée.
Received : 2014-04-28
Revised : 2016-01-08
Accepted : 2016-06-14
Published online : 2017-01-10
Classification:  20E08,  20F65,  20E32
Keywords: Almost automorphisms of trees, Neretin group, compact presentability, Dehn function
@article{AIF_2017__67_1_329_0,
     author = {Le Boudec, Adrien},
     title = {Compact presentability of tree almost automorphism groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {1},
     year = {2017},
     pages = {329-365},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_1_329_0}
}
Compact presentability of tree almost automorphism groups. Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 329-365. https://aif.centre-mersenne.org/item/AIF_2017__67_1_329_0/

[1] Bader, Uri; Caprace, Pierre-Emmanuel; Gelander, Tsachik; Mozes, Shahar Simple groups without lattices, Bull. Lond. Math. Soc., Tome 44 (2012) no. 1, pp. 55-67

[2] Barnea, Yiftach; Ershov, Mikhail; Weigel, Thomas Abstract commensurators of profinite groups, Trans. Amer. Math. Soc., Tome 363 (2011) no. 10, pp. 5381-5417

[3] Bartholdi, Laurent; Grigorchuk, Rostislav On parabolic subgroups and Hecke algebras of some fractal groups, Serdica Math. J., Tome 28 (2002) no. 1, pp. 47-90

[4] Bartholdi, Laurent; Grigorchuk, Rostislav; Šuniḱ, Zoran Branch groups, Handbook of algebra, Vol. 3, North-Holland, Amsterdam (2003), pp. 989-1112

[5] Birget, Jean-Camille The groups of Richard Thompson and complexity, Internat. J. Algebra Comput., Tome 14 (2004) no. 5-6, pp. 569-626 (International Conference on Semigroups and Groups in honor of the 65th birthday of Prof. John Rhodes)

[6] Bourbaki, Nicolas Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris (1971)

[7] Bowditch, Brian H. A short proof that a subquadratic isoperimetric inequality implies a linear one, Michigan Math. J., Tome 42 (1995) no. 1, pp. 103-107

[8] Brown, Kenneth S. Finiteness properties of groups, J. Pure Appl. Algebra, Tome 44 (1987) no. 1-3, pp. 45-75 (Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985))

[9] Burillo, José Quasi-isometrically embedded subgroups of Thompson’s group F, J. Algebra, Tome 212 (1999) no. 1, pp. 65-78

[10] Burillo, José; Cleary, Sean; Stein, Melanie I. Metrics and embeddings of generalizations of Thompson’s group F, Trans. Amer. Math. Soc., Tome 353 (2001) no. 4, pp. 1677-1689

[11] Burillo, José; Cleary, Sean; Stein, Melanie I.; Taback, Jennifer Combinatorial and metric properties of Thompson’s group T, Trans. Amer. Math. Soc., Tome 361 (2009) no. 2, pp. 631-652

[12] Cannon, James W.; Floyd, William J.; Parry, Walter R. Introductory notes on Richard Thompson’s groups, Enseign. Math. (2), Tome 42 (1996) no. 3-4, pp. 215-256

[13] Caprace, Pierre-Emmanuel; De Medts, Tom Simple locally compact groups acting on trees and their germs of automorphisms, Transform. Groups, Tome 16 (2011) no. 2, pp. 375-411

[14] Cornulier, Yves; De La Harpe, Pierre Metric geometry of locally compact groups (2015) (Book in preparation, http://arxiv.org/abs/1403.3796v3)

[15] Elder, Murray; Willis, Georges Totally disconnected groups from Baumslag-Solitar groups (2013) (http://arxiv.org/abs/1301.4775v3 )

[16] Grigorchuk, Rostislav On Burnside’s problem on periodic groups, Funktsional. Anal. i Prilozhen., Tome 14 (1980) no. 1, p. 53-54

[17] Grigorchuk, Rostislav Solved and unsolved problems around one group, Infinite groups: geometric, combinatorial and dynamical aspects, Birkhäuser, Basel (Progr. Math.) Tome 248 (2005), pp. 117-218

[18] Groves, John; Hermiller, Susan Isoperimetric inequalities for soluble groups, Geom. Dedicata, Tome 88 (2001) no. 1-3, pp. 239-254

[19] Guba, Victor S. Polynomial isoperimetric inequalities for Richard Thompson’s groups F, T, and V, Algorithmic problems in groups and semigroups (Lincoln, NE, 1998), Birkhäuser Boston, Boston, MA (Trends Math.) (2000), pp. 91-120

[20] Guba, Victor S. The Dehn function of Richard Thompson’s group F is quadratic, Invent. Math., Tome 163 (2006) no. 2, pp. 313-342

[21] Herman, Michael R. Simplicité du groupe des difféomorphismes de classe C , isotopes à l’identité, du tore de dimension n, C. R. Acad. Sci. Paris Sér. A-B, Tome 273 (1971), p. A232-A234

[22] Higman, Graham Finitely presented infinite simple groups, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra (1974), vii+82 pages (Notes on Pure Mathematics, No. 8 (1974))

[23] Kapoudjian, Christophe Simplicity of Neretin’s group of spheromorphisms, Ann. Inst. Fourier (Grenoble), Tome 49 (1999) no. 4, pp. 1225-1240

[24] Nekrashevych, Volodymyr Cuntz-Pimsner algebras of group actions, J. Operator Theory, Tome 52 (2004) no. 2, pp. 223-249

[25] Nekrashevych, Volodymyr Self-similar groups, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 117 (2005), xii+231 pages

[26] Nekrashevych, Volodymyr Finitely presented groups associated with expanding maps (2013) (http://arxiv.org/abs/1312.5654v1 )

[27] Neretin, Yuri A. On Combinatorial analogus of the group of diffeomorphisms of the circle, Izv. Ross. Akad. Nauk Ser. Mat., Tome 56 (1992) no. 5, pp. 1072-1085

[28] Reid, Colin D. The profinite completion of a group localised at a subgroup (http://arxiv.org/abs/1208.3554v2 )

[29] Röver, Claas E. Constructing finitely presented simple groups that contain Grigorchuk groups, J. Algebra, Tome 220 (1999) no. 1, pp. 284-313

[30] Schlichting, Günter Operationen mit periodischen Stabilisatoren, Arch. Math., Tome 34 (1980) no. 2, pp. 97-99

[31] Shalom, Yehuda; Willis, George A. Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity, Geom. Funct. Anal., Tome 23 (2013) no. 5, pp. 1631-1683

[32] Šunić, Zoran Hausdorff dimension in a family of self-similar groups, Geom. Dedicata, Tome 124 (2007), pp. 213-236

[33] Tzanev, Kroum Hecke C * -algebras and amenability, J. Operator Theory, Tome 50 (2003) no. 1, pp. 169-178