# ANNALES DE L'INSTITUT FOURIER

${C}^{*}$-simplicity of HNN extensions and groups acting on trees
Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1497-1543.

We study non-ascending HNN extensions acting on their Bass–Serre tree and characterize ${C}^{*}$-simplicity and the unique trace property by means of the kernel and quasi-kernels of the HNN extension in question. We also present a concrete example of an HNN extension that is a new example of a group that is not ${C}^{*}$-simple but does have the unique trace property. Additionally, we include certain more general results, mostly based on previous work of various authors, concerning ${C}^{*}$-simplicity of groups admitting extreme boundary actions, and in particular, groups acting on trees.

Nous étudions les extensions HNN non-ascendantes agissant sur leur arbre de Bass–Serre and caractérisons la ${C}^{*}$-simplicité et la propriété d’unicité de la trace à l’aide du noyau et des quasi-noyaux des extensions HNN en question. Nous présentons aussi un exemple concret d’extension HNN fournissant un nouvel exemple de groupe ${C}^{*}$-simple mais ne possédant pas la propriété d’unicité de la trace. De plus, nous obtenons certains resultats plus généraux, la plupart se fondant sur des travaux antérieurs d’auteurs divers, concernant la ${C}^{*}$-simplicité de groupes admettant des actions extrêmes sur des bords, et, en particulier, de groupes agissant sur des arbres.

Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3378
Classification: 22D25, 20E06, 46L05, 20E08
Keywords: $C^{*}$-simplicity, HNN extension.
Bryder, Rasmus Sylvester 1; Ivanov, Nikolay A. 2; Omland, Tron 3, 4

1 Department of Mathematics University of Copenhagen Universitetsparken 5 2100 Copenhagen (Denmark)
2 Faculty of Mathematics and Informatics University of Sofia blvd. James Bourchier 5 BG-1164 Sofia (Bulgaria)
3 Department of Computer Science Oslo Metropolitan University NO-0130 Oslo (Norway)
4 Department of Mathematics University of Oslo NO-0316 Oslo (Norway)
@article{AIF_2020__70_4_1497_0,
author = {Bryder, Rasmus Sylvester and Ivanov, Nikolay A. and Omland, Tron},
title = {$C^{*}$-simplicity of {HNN~extensions} and groups~acting~on~trees},
journal = {Annales de l'Institut Fourier},
pages = {1497--1543},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {70},
number = {4},
year = {2020},
doi = {10.5802/aif.3378},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3378/}
}
TY  - JOUR
AU  - Bryder, Rasmus Sylvester
AU  - Ivanov, Nikolay A.
AU  - Omland, Tron
TI  - $C^{*}$-simplicity of HNN extensions and groups acting on trees
JO  - Annales de l'Institut Fourier
PY  - 2020
SP  - 1497
EP  - 1543
VL  - 70
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3378/
DO  - 10.5802/aif.3378
LA  - en
ID  - AIF_2020__70_4_1497_0
ER  -
%0 Journal Article
%A Bryder, Rasmus Sylvester
%A Ivanov, Nikolay A.
%A Omland, Tron
%T $C^{*}$-simplicity of HNN extensions and groups acting on trees
%J Annales de l'Institut Fourier
%D 2020
%P 1497-1543
%V 70
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3378/
%R 10.5802/aif.3378
%G en
%F AIF_2020__70_4_1497_0
Bryder, Rasmus Sylvester; Ivanov, Nikolay A.; Omland, Tron. $C^{*}$-simplicity of HNN extensions and groups acting on trees. Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1497-1543. doi : 10.5802/aif.3378. https://aif.centre-mersenne.org/articles/10.5802/aif.3378/

[1] Alperin, Roger; Bass, Hyman Length functions of group actions on $\Lambda$-trees, Combinatorial group theory and topology (Annals of Mathematics Studies), Volume 111, Princeton University Press, 1987, pp. 265-378 | DOI | MR | Zbl

[2] Bass, Hyman Covering theory for graphs of groups, J. Pure Appl. Algebra, Volume 89 (1993) no. 1-2, pp. 3-47 | DOI | MR | Zbl

[3] Bogopolski, Oleg Introduction to Group Theory, European Mathematical Society, 2008 | DOI | MR | Zbl

[4] Breuillard, Emmanuel; Kennedy, Matthew; Kalantar, Mehrdad; Ozawa, Narutaka ${C}^{*}$-simplicity and the unique trace property for discrete groups, Publ. Math., Inst. Hautes Étud. Sci. (2017) no. 126, pp. 35-71 | DOI | MR | Zbl

[5] Brown, Nathaniel P.; Ozawa, Narutaka ${C}^{*}$-algebras and Finite-Dimensional Approximations, Grad. Stud. Math., 88, American Mathematical Society, 2008 | MR | Zbl

[6] Brownlowe, Nathan; Mundey, Alexander; Pask, David; Spielberg, Jack; Thomas, Anne ${C}^{*}$-algebras associated to graphs of groups, Adv. Math. (2017) no. 316, pp. 114-186 | DOI | MR | Zbl

[7] de Cornulier, Yves Infinite conjugacy classes in groups acting on trees, Groups Geom. Dyn., Volume 3 (2009) no. 2, pp. 267-277 | DOI | MR | Zbl

[8] Culler, Marc; Morgan, John W. Group actions on $ℝ$-trees, Proc. Lond. Math. Soc., Volume 55 (1987) no. 3, pp. 571-604 | DOI | Zbl

[9] Day, Mahlon M. Amenable semigroups, Ill. J. Math., Volume 1 (1957), pp. 509-544 | MR | Zbl

[10] Furman, Alex On minimal strongly proximal actions of locally compact groups, Isr. J. Math., Volume 136 (2003), pp. 173-187 | DOI | MR | Zbl

[11] Furstenberg, Harry Boundary theory and stochastic processes on homogeneous spaces, Harmonic Analysis on Homogeneous Spaces, American Mathematical Society, 1973, pp. 193-229 | DOI | Zbl

[12] Glasner, Shmuel Topological dynamics and group theory, Trans. Am. Math. Soc., Volume 187 (1974), pp. 327-334 | DOI | MR | Zbl

[13] Haagerup, Uffe A new look at ${C}^{*}$-simplicity and the unique trace property of a group, Operator algebras and applications, Volume 12 (2016), pp. 161-170 | Zbl

[14] Haagerup, Uffe; Olesen, Kristian Knudsen Non-inner amenability of the Thompson groups $T$ and $V$, J. Funct. Anal., Volume 272 (2017) no. 11, pp. 4838-4852 | DOI | MR | Zbl

[15] Halmos, Paul R. Lectures on Boolean algebras, Springer, 1974 | Zbl

[16] de la Harpe, Pierre Reduced ${C}^{*}$-algebras of discrete groups which are simple with unique trace, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) (Lecture Notes in Mathematics), Volume 1132, Springer, 1985, pp. 230-253 | DOI | MR | Zbl

[17] de la Harpe, Pierre On simplicity of reduced ${C}^{*}$-algebras of groups, Bull. Lond. Math. Soc., Volume 39 (2007) no. 1, pp. 1-26 | DOI | MR | Zbl

[18] de la Harpe, Pierre; Préaux, Jean-Philippe ${C}^{*}$-simple groups: amalgamated free products, HNN extensions, and fundamental groups of $3$-manifolds, J. Topol. Anal., Volume 3 (2011) no. 4, pp. 451-489 | DOI | MR | Zbl

[19] Higman, Graham; Neumann, Bernhard H.; Neumann, Hanna Embedding theorems for groups, J. Lond. Math. Soc., Volume 24 (1949), pp. 247-254 | DOI | MR | Zbl

[20] Ivanov, Nikolay A. On the structure of some reduced amalgamated free product ${C}^{*}$-algebras, Int. J. Math., Volume 22 (2011) no. 2, pp. 281-306 | DOI | MR | Zbl

[21] Ivanov, Nikolay A. Two families of examples of groups acting on trees with nontrivial quasi-kernels (2020) (https://arxiv.org/abs/2003.10196)

[22] Ivanov, Nikolay A.; Omland, Tron ${C}^{*}$-simplicity of free products with amalgamation and radical classes of groups, J. Funct. Anal., Volume 272 (2017) no. 9, pp. 3712-3741 | DOI | MR | Zbl

[23] Kalantar, Mehrdad; Kennedy, Matthew Boundaries of reduced ${C}^{*}$-algebras of discrete groups, J. Reine Angew. Math., Volume 727 (2017), pp. 247-267 | MR | Zbl

[24] Kennedy, Matthew An intrinsic characterization of ${C}^{*}$-simplicity (2018) To appear in Ann. Sci. Éc. Norm. Supér. (4)

[25] Laca, Marcelo; Spielberg, Jack Purely infinite ${C}^{*}$-algebras from boundary actions of discrete groups, J. Reine Angew. Math., Volume 480 (1996), pp. 125-139 | MR | Zbl

[26] Le Boudec, Adrien ${C}^{*}$-simplicity and the amenable radical, Invent. Math., Volume 209 (2017) no. 1, pp. 159-174 | DOI | MR | Zbl

[27] Le Boudec, Adrien; Matte Bon, Nicolás Subgroup dynamics and ${C}^{*}$-simplicity of groups of homeomorphisms, Ann. Sci. Éc. Norm. Supér., Volume 51 (2018) no. 3, pp. 557-602 | DOI | MR | Zbl

[28] Monod, Nicolas; Shalom, Yehuda Cocycle superrigidity and bounded cohomology for negatively curved spaces, J. Differ. Geom., Volume 67 (2004) no. 3, pp. 395-455 | MR | Zbl

[29] Ozawa, Narutaka Lecture on the Furstenberg boundary and ${C}^{*}$-simplicity (2014) (http://www.kurims.kyoto-u.ac.jp/~narutaka/notes/yokou2014.pdf)

[30] Paschke, William L.; Salinas, Norberto ${C}^{*}$-algebras associated with free products of groups, Pac. J. Math., Volume 82 (1979) no. 1, pp. 211-221 | DOI | MR | Zbl

[31] Powers, Robert T. Simplicity of the ${C}^{*}$-algebra associated with the free group on two generators, Duke Math. J., Volume 42 (1975) no. 1, pp. 151-156 | MR | Zbl

[32] Serre, Jean-Pierre Trees, Springer, 1980 | Zbl

[33] Sidki, Said; Wilson, John S. Free subgroups of branch groups, Arch. Math., Volume 80 (2003) no. 5, pp. 458-463 | DOI | MR | Zbl

[34] Tu, Jean-Louis La conjecture de Baum–Connes pour les feuilletages moyennables, $K$-Theory, Volume 17 (1999) no. 3, pp. 215-264 | MR | Zbl

Cited by Sources: