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(Self-)similar groups and the Farrell-Jones conjectures. (English) Zbl 1293.20026

Summary: We show that contracting self-similar groups satisfy the Farrel-Jones conjectures as soon as their universal contracting cover is non-positively curved. This applies in particular to bounded self-similar groups.
We define, along the way, a general notion of contraction for groups acting on a rooted tree in a not necessarily self-similar manner.

MSC:

20E08 Groups acting on trees
19D10 Algebraic \(K\)-theory of spaces
20F65 Geometric group theory
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
19A31 \(K_0\) of group rings and orders
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[1] S. V. Aleshin, Finite automata and Burnside’s problem for periodic groups. Mat. Zametki 11 (1972), 319-328; English transl. Math. Notes 11 (1972), 199-203. · Zbl 0253.20049 · doi:10.1007/BF01098526
[2] S. V. Aleshin, A free group of finite automata. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1983 (1983), No. 4, 12-14; English transl. Moscow Univ. Math. Bull. 38 (1983), No. 4, 10-13. · Zbl 0513.68044
[3] G. Amir, O. Angel, and B. Virag, Amenability of linear-activity automaton groups. J. Eur. Math. Soc. ( JEMS), to appear; Preprint 2009.
[4] A. Bartels, F. T. Farrell, and W. Lück, The Farrell-Jones Conjecture for cocompact lattices in virtually connected Lie groups. Preprint 2011. · Zbl 1307.18012
[5] A. Bartels and W. Lück, On crossed product rings with twisted involutions, their mod- ule categories and L-theory. In Cohomology of groups and algebraic K -theory, Adv. Lect. Math. (ALM) 12, Internat. Press, Somerville, MA, 2010, 1-54. · Zbl 1205.19006
[6] A. Bartels and W. Lück, The Borel conjecture for hyperbolic and CAT.0/ groups. Ann. of Math. (2) 175 (2012), 631-689. · Zbl 1256.57021 · doi:10.4007/annals.2012.175.2.5
[7] A. Bartels, W. Lück, and H. Reich, On the Farrell-Jones conjecture and its applications. J. Topol. 1 (2008), 57-86. · Zbl 1141.19002 · doi:10.1112/jtopol/jtm008
[8] L. Bartholdi, R. I. Grigorchuk, and Z. Šuniḱ, Branch groups. In Handbook of algebra . Vol. 3, North-Holland, Amsterdam 2003, 989-1112. · Zbl 1140.20306
[9] L. Bartholdi, V. A. Kaimanovich, and V. V. Nekrashevych, On amenability of automata groups. Duke Math. J. 154 (2010), 575-598. · Zbl 1268.20026 · doi:10.1215/00127094-2010-046
[10] L. Bartholdi and V. V. Nekrashevych, Iterated monodromy groups of quadratic polyno- mials. I. Groups Geom. Dyn. 2 (2008), 309-336. · Zbl 1153.37379 · doi:10.4171/GGD/42
[11] L. Bartholdi and B. Virág, Amenability via random walks. Duke Math. J. 130 (2005), 39-56. · Zbl 1104.43002 · doi:10.1215/S0012-7094-05-13012-5
[12] E. Bondarenko and V. Nekrashevych, Post-critically finite self-similar groups. Algebra Discrete Math. 2003 (2003), no. 4, 21-32. · Zbl 1068.20028
[13] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature . Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin 1999. · Zbl 0988.53001
[14] X. Buff, A. Epstein, S. Koch, and K. Pilgrim, On Thurston’s pullback map. In Complex dy- namics. Families and friends .A K Peters, Wellesley, MA, 2009, 561-583. · Zbl 1180.37065
[15] M. Burger and S. Mozes, Finitely presented simple groups and products of trees. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 747-752. · Zbl 0966.20013 · doi:10.1016/S0764-4442(97)86938-8
[16] M. Burger and S. Mozes, Groups acting on trees: from local to global structure. Inst. Hautes Études Sci. Publ. Math. 92 (2000), 113-150. · Zbl 1007.22012 · doi:10.1007/BF02698915
[17] E. S. Golod, On nil-algebras and finitely approximable p-groups. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 273-276; English transl. Amer. Math. Soc. Transl. (2) 48, Amer. Math. Soc., Providence, RI, 1965, 103-106. · Zbl 0215.39202
[18] R. I. Grigorchuk, On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR 271 (1983), 30-33; English transl. Soviet Math. Dokl. 28 (1983), 23-26. · Zbl 0547.20025
[19] R. I. Grigorchuk, Degrees of growth of finitely generated groups, and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), 939-985; English transl. Math. USSR-Izv. f25 (1985), 259-300. · Zbl 0583.20023 · doi:10.1070/IM1985v025n02ABEH001281
[20] R. I. Grigorchuk, An example of a finitely presented amenable group not belonging to the class EG. Mat. Sb. 189 (1998), 79-100; English transl. Sb. Math. 189 (1998), 75-95. · Zbl 0931.43003 · doi:10.1070/SM1998v189n01ABEH000293
[21] R. I. Grigorchuk and A. \? Zuk, Spectral properties of a torsion-free weakly branch group defined by a three state automaton. In Computational and statistical group theory (Las Vegas, NV/Hoboken, NJ, 2001), Contemp. Math. 298, Amer. Math. Soc., Providence, RI, 2002, 57-82. · Zbl 1057.60045
[22] N. Gupta and S. Sidki, On the Burnside problem for periodic groups. Math. Z. 182 (1983), 385-388. · Zbl 0513.20024 · doi:10.1007/BF01179757
[23] L. Kaloujnine and M. Krasner, Le produit complet des groupes de permutations et le problème d’extension des groupes. C. R. Acad. Sci. Paris 227 (1948), 806-808. · Zbl 0038.16203
[24] S. C. Koch, Teichmüller theory and applications to endomorphisms of P n. Ph.D. thesis, Université de Provence, Marseille 2007.
[25] V. Nekrashevych, Self-similar groups . Math. Surveys Monogr. 117, Amer. Math. Soc., Providence, RI, 2005. · Zbl 1087.20032
[26] S. Sidki, Finite automata of polynomial growth do not generate a free group. Geom. Dedicata 108 (2004), 193-204. · Zbl 1075.20011 · doi:10.1007/s10711-004-2368-0
[27] V. xI. Suš\check cans0ki\?ı, Periodic p-groups of permutations and the unrestricted Burnside prob- lem. Dokl. Akad. Nauk SSSR 247 (1979), 557-561; English transl. Soviet Math. Dokl. 20 (1979), 766-770. · Zbl 0428.20023
[28] C. Wegner, The K-theoretic Farrell-Jones conjecture for CAT.0/-groups. Proc. Amer. Math. Soc. 140 (2012), 779-793. · Zbl 1240.19003 · doi:10.1090/S0002-9939-2011-11150-X
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