Boundary 𝐶*-algebras for acylindrical groups

Robertson, Guyan

doi:10.1090/S0002-9939-08-09453-7

Boundary $C^*$-algebras for acylindrical groups
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by Guyan Robertson

Proc. Amer. Math. Soc. 136 (2008), 3851-3860

DOI: https://doi.org/10.1090/S0002-9939-08-09453-7

Published electronically: June 3, 2008

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Abstract:

Let $\Delta$ be an infinite, locally finite tree with more than two ends. Let $\Gamma <\operatorname {Aut}(\Delta )$ be an acylindrical uniform lattice. Then the boundary algebra $\mathcal {A}_\Gamma = C(\partial \Delta )\rtimes \Gamma$ is a simple Cuntz-Krieger algebra whose K-theory is determined explicitly.

References

Claire Anantharaman-Delaroche, Systèmes dynamiques non commutatifs et moyennabilité, Math. Ann. 279 (1987), no. 2, 297–315 (French). MR 919508, DOI 10.1007/BF01461725
Anne Broise-Alamichel and Frédéric Paulin, Sur le codage du flot géodésique dans un arbre, Ann. Fac. Sci. Toulouse Math. (6) 16 (2007), no. 3, 477–527 (French, with English and French summaries). MR 2379050, DOI 10.5802/afst.1157
Joachim Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185. MR 467330, DOI 10.1007/BF01625776
Joachim Cuntz, $K$-theory for certain $C^{\ast }$-algebras, Ann. of Math. (2) 113 (1981), no. 1, 181–197. MR 604046, DOI 10.2307/1971137
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
Su Shing Chen, Limit sets of automorphism groups of a tree, Proc. Amer. Math. Soc. 83 (1981), no. 2, 437–441. MR 624949, DOI 10.1090/S0002-9939-1981-0624949-8
Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
Eberhard Kirchberg, Exact $\textrm {C}^*$-algebras, tensor products, and the classification of purely infinite algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 943–954. MR 1403994
Guyan Robertson, Boundary actions for affine buildings and higher rank Cuntz-Krieger algebras, $C^*$-algebras (Münster, 1999) Springer, Berlin, 2000, pp. 182–202. MR 1798597
Guyan Robertson, Boundary operator algebras for free uniform tree lattices, Houston J. Math. 31 (2005), no. 3, 913–935. MR 2148805
Guyan Robertson and Tim Steger, Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras, J. Reine Angew. Math. 513 (1999), 115–144. MR 1713322, DOI 10.1515/crll.1999.057
Z. Sela, Acylindrical accessibility for groups, Invent. Math. 129 (1997), no. 3, 527–565. MR 1465334, DOI 10.1007/s002220050172
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7