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Amenability of Discrete Groups by Examples
About this Title
Kate Juschenko, University of Texas, Austin, TX
Publication: Mathematical Surveys and Monographs
Publication Year:
2022; Volume 266
ISBNs: 978-1-4704-7032-6 (print); 978-1-4704-7109-5 (online)
DOI: https://doi.org/10.1090/surv/266
Table of Contents
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Front/Back Matter
Chapters
- Introduction
- The first germs of amenability: Paradoxical decompositions
- Elementary amenable groups
- The topological full group of Cantor minimal system
- Lamplighter actions and extensive amenability
- Amenability of topological full groups
- Subgroups of topological full groups of intermediate growth
- An amenability criterion via actions
- Groups acting on Bratteli diagrams
- Groups acting on rooted trees
- Appendix A. Definitions of amenability and basic facts
- Appendix B. Related open problems
- S. I. Adyan, Random walks on free periodic groups, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1139–1149, 1343 (Russian). MR 682486
- S. V. Alešin, Finite automata and the Burnside problem for periodic groups, Mat. Zametki 11 (1972), 319–328 (Russian). MR 301107
- Gideon Amir and Bálint Virág, Positive speed for high-degree automaton groups, Groups Geom. Dyn. 8 (2014), no. 1, 23–38. MR 3209701, DOI 10.4171/GGD/215
- Gideon Amir and Bálint Virág, Speed exponents of random walks on groups, Int. Math. Res. Not. IMRN 9 (2017), 2567–2598. MR 3658209, DOI 10.1093/imrn/rnv378
- Gideon Amir, Omer Angel, and Bálint Virág, Amenability of linear-activity automaton groups, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 705–730. MR 3085088, DOI 10.4171/JEMS/373
- S. Banach, Théorie des opérations linéaires, Instytut Matematyczny Polskiej Akademi Nauk, 1932.
- Stefan Banach, Théorie des opérations linéaires, Chelsea Publishing Co., New York, 1955 (French). MR 0071726
- S. Banach, Sur le probléme de la mesure, Fund. Math. 4, 7–33 (1923).
- S. Banach and A. Tarski, Sur la décomposition des ensembles de points en parties respectivement congruents, Fund. Math., 14 (1929), 127–131.
- Laurent Bartholdi and Anna Erschler, Growth of permutational extensions, Invent. Math. 189 (2012), no. 2, 431–455. MR 2947548, DOI 10.1007/s00222-011-0368-x
- Laurent Bartholdi and Anna Erschler, Poisson-Furstenberg boundary and growth of groups, Probab. Theory Related Fields 168 (2017), no. 1-2, 347–372. MR 3651055, DOI 10.1007/s00440-016-0712-6
- Laurent Bartholdi, Vadim A. Kaimanovich, and Volodymyr V. Nekrashevych, On amenability of automata groups, Duke Math. J. 154 (2010), no. 3, 575–598. MR 2730578, DOI 10.1215/00127094-2010-046
- Laurent Bartholdi and Bálint Virág, Amenability via random walks, Duke Math. J. 130 (2005), no. 1, 39–56. MR 2176547, DOI 10.1215/S0012-7094-05-13012-5
- Laurent Bartholdi, Rostislav Grigorchuk, and Volodymyr Nekrashevych, From fractal groups to fractal sets, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 25–118. MR 2091700
- Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR 2415834, DOI 10.1017/CBO9780511542749
- G. Bell and A. Dranishnikov, Asymptotic dimension, Topology Appl. 155 (2008), no. 12, 1265–1296. MR 2423966, DOI 10.1016/j.topol.2008.02.011
- Itai Benjamini and Christopher Hoffman, $\omega$-periodic graphs, Electron. J. Combin. 12 (2005), Research Paper 46, 12. MR 2176522, DOI 10.37236/1943
- I. Benjamini and O. Schramm, Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant, Geom. Funct. Anal. 7 (1997), no. 3, 403–419. MR 1466332, DOI 10.1007/PL00001625
- Jean Bellissard, Antoine Julien, and Jean Savinien, Tiling groupoids and Bratteli diagrams, Ann. Henri Poincaré 11 (2010), no. 1-2, 69–99. MR 2658985, DOI 10.1007/s00023-010-0034-7
- V. Bergelson, Questions on amenability, L’Enseignement Mathématique 2.54 (2008): 28–30.
- S. Bezuglyi and K. Medynets, Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems, Colloq. Math. 110 (2008), no. 2, 409–429. MR 2353913, DOI 10.4064/cm110-2-6
- Bruce Blackadar, $K$-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031
- C. Bleak and K. Juschenko, Ideal structure of the C*-algebra of Thompson group T, arXiv preprint arXiv:1409.8099, 2014.
- N. Bogolyubov, On some ergodic properties of continuous groups of transformations, 1939. Published first in Ukrainian (in Sci- entific Notes of Kiev State University of T.G. Shevchenko, Physics - Mathematics zbirnyk, 4, N. 5 (1939), 45–52), and then in Russian (In “Selected works”, Vol. 1, Kiev 1969, pp. 561–569).
- Nicolas Kryloff and Nicolas Bogoliouboff, La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire, Ann. of Math. (2) 38 (1937), no. 1, 65–113 (French). MR 1503326, DOI 10.2307/1968511
- Ievgen Bondarenko, Groups generated by bounded automata and their Schreier graphs, ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–Texas A&M University. MR 2711289
- Ievgen V. Bondarenko, Finite generation of iterated wreath products, Arch. Math. (Basel) 95 (2010), no. 4, 301–308. MR 2727305, DOI 10.1007/s00013-010-0169-2
- Ievgen Bondarenko, Tullio Ceccherini-Silberstein, Alfredo Donno, and Volodymyr Nekrashevych, On a family of Schreier graphs of intermediate growth associated with a self-similar group, European J. Combin. 33 (2012), no. 7, 1408–1421. MR 2923458, DOI 10.1016/j.ejc.2012.03.006
- Ievgen V. Bondarenko and Dmytro M. Savchuk, On Sushchansky $p$-groups, Algebra Discrete Math. 2 (2007), 22–42. MR 2364061
- Mike Boyle and Jun Tomiyama, Bounded topological orbit equivalence and $C^*$-algebras, J. Math. Soc. Japan 50 (1998), no. 2, 317–329. MR 1613140, DOI 10.2969/jmsj/05020317
- Ola Bratteli, Inductive limits of finite dimensional $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234. MR 312282, DOI 10.1090/S0002-9947-1972-0312282-2
- Jérémie Brieussel, Amenability and non-uniform growth of some directed automorphism groups of a rooted tree, Math. Z. 263 (2009), no. 2, 265–293. MR 2534118, DOI 10.1007/s00209-008-0417-3
- Jérémie Brieussel, Folner sets of alternate directed groups, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 3, 1109–1130 (English, with English and French summaries). MR 3330165
- Tullio Ceccherini-Silberstein and Michel Coornaert, Cellular automata and groups, Computational complexity. Vols. 1–6, Springer, New York, 2012, pp. 336–349. MR 3074498, DOI 10.1007/978-1-4614-1800-9_{2}3
- P. de lya Arp, R. I. Grigorchuk, and T. Chekerini-Sil′berstaĭn, Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces, Tr. Mat. Inst. Steklova 224 (1999), no. Algebra. Topol. Differ. Uravn. i ikh Prilozh., 68–111 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 1(224) (1999), 57–97. MR 1721355
- Maksym Chornyi, Kate Juschenko, and Volodymyr Nekrashevych, On topological full groups of $\Bbb Z^d$-actions, Groups Geom. Dyn. 14 (2020), no. 1, 61–79. MR 4077654, DOI 10.4171/ggd/534
- Ching Chou, Elementary amenable groups, Illinois J. Math. 24 (1980), no. 3, 396–407. MR 573475
- A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
- A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), no. 4, 431–450 (1982). MR 662736, DOI 10.1017/s014338570000136x
- I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433, DOI 10.1007/978-1-4615-6927-5
- Yves de Cornulier, Groupes pleins-topologiques (d’après Matui, Juschenko, Monod, $\ldots$), Astérisque 361 (2014), Exp. No. 1064, viii, 183–223 (French, with French summary). MR 3289281
- Yves Cornulier and Pierre de la Harpe, Metric geometry of locally compact groups, EMS Tracts in Mathematics, vol. 25, European Mathematical Society (EMS), Zürich, 2016. Winner of the 2016 EMS Monograph Award. MR 3561300, DOI 10.4171/166
- François Dahmani, Koji Fujiwara, and Vincent Guirardel, Free groups of interval exchange transformations are rare, Groups Geom. Dyn. 7 (2013), no. 4, 883–910. MR 3134029, DOI 10.4171/GGD/209
- Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. MR 92128
- Mahlon M. Day, Semigroups and amenability, Semigroups (Proc. Sympos., Wayne State Univ., Detroit, Mich., 1968) Academic Press, New York, 1969, pp. 5–53. MR 0265502
- W. A. Deuber, M. Simonovits, and V. T. Sós, A note on paradoxical metric spaces, Studia Sci. Math. Hungar. 30 (1995), no. 1-2, 17–23. MR 1341564
- Jacques Dixmier, Les $C^*$-algèbres et leurs représentations, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Paris, 1996 (French). Reprint of the second (1969) edition. MR 1452364
- Jacques Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (Algèbres de von Neumann), Cahiers Scientifiques, Fasc. XXV, Gauthier-Villars, Paris, 1957 (French). MR 0094722
- Eric K. van Douwen, Measures invariant under actions of $F_2$, Topology Appl. 34 (1990), no. 1, 53–68. MR 1035460, DOI 10.1016/0166-8641(90)90089-K
- David Damanik and Daniel Lenz, Substitution dynamical systems: characterization of linear repetitivity and applications, J. Math. Anal. Appl. 321 (2006), no. 2, 766–780. MR 2241154, DOI 10.1016/j.jmaa.2005.09.004
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Gábor Elek and Nicolas Monod, On the topological full group of a minimal Cantor $\mathbf {Z}^2$-system, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3549–3552. MR 3080176, DOI 10.1090/S0002-9939-2013-11654-0
- R. Exel and J. Renault, $AF$-algebras and the tail-equivalence relation on Bratteli diagrams, Proc. Amer. Math. Soc. 134 (2006), no. 1, 193–206. MR 2170559, DOI 10.1090/S0002-9939-05-08129-3
- Mikhail Ershov, Golod-Shafarevich groups with property $(T)$ and Kac-Moody groups, Duke Math. J. 145 (2008), no. 2, 309–339. MR 2449949, DOI 10.1215/00127094-2008-053
- Mikhail Ershov, Golod-Shafarevich groups: a survey, Internat. J. Algebra Comput. 22 (2012), no. 5, 1230001, 68. MR 2949205, DOI 10.1142/S0218196712300010
- Mikhail Ershov, Gili Golan, and Mark Sapir, The Tarski numbers of groups, Adv. Math. 284 (2015), 21–53. MR 3391070, DOI 10.1016/j.aim.2015.07.010
- Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
- Elisabeth Fink, A finitely generated branch group of exponential growth without free subgroups, J. Algebra 397 (2014), 625–642. MR 3119242, DOI 10.1016/j.jalgebra.2013.06.030
- T. Gelander, Lecture notes: analytic group theory, https://www.weizmann.ac.il/math/Gelander/sites/math.Gelander/files/uploads/AGT.pdf.
- Étienne Ghys and Yves Carrière, Relations d’équivalence moyennables sur les groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 19, 677–680 (French, with English summary). MR 802650
- Thierry Giordano, Ian F. Putnam, and Christian F. Skau, Full groups of Cantor minimal systems, Israel J. Math. 111 (1999), 285–320. MR 1710743, DOI 10.1007/BF02810689
- Eli Glasner and Benjamin Weiss, Weak orbit equivalence of Cantor minimal systems, Internat. J. Math. 6 (1995), no. 4, 559–579. MR 1339645, DOI 10.1142/S0129167X95000213
- E. S. Golod and I. R. Šafarevič, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272 (Russian). MR 0161852
- F. P. Greenleaf, Amenable actions of locally compact groups, J. Functional Analysis 4 (1969), 295–315. MR 0246999, DOI 10.1016/0022-1236(69)90016-0
- R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskiĭ, Automata, dynamical systems, and groups, Tr. Mat. Inst. Steklova 231 (2000), no. Din. Sist., Avtom. i Beskon. Gruppy, 134–214 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 4(231) (2000), 128–203. MR 1841755
- R. I. Grigorčuk, On Burnside’s problem on periodic groups, Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 53–54 (Russian). MR 565099
- R. I. Grigorchuk, Symmetrical random walks on discrete groups, Multicomponent random systems, Adv. Probab. Related Topics, vol. 6, Dekker, New York, 1980, pp. 285–325. MR 599539
- R. Grigorchuk, Milnor’s problem on the growth of groups, Sov. Math., Dokl, 28 (1983), 23–26.
- R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 5, 939–985 (Russian). MR 764305
- R. I. Grigorchuk, An example of a finitely presented amenable group that does not belong to the class EG, Mat. Sb. 189 (1998), no. 1, 79–100 (Russian, with Russian summary); English transl., Sb. Math. 189 (1998), no. 1-2, 75–95. MR 1616436, DOI 10.1070/SM1998v189n01ABEH000293
- R. I. Grigorchuk, Superamenability and the occurrence problem of free semigroups, Funktsional. Anal. i Prilozhen. 21 (1987), no. 1, 74–75 (Russian). MR 888020
- Rostislav Grigorchuk and Pierre de la Harpe, Amenability and ergodic properties of topological groups: from Bogolyubov onwards, Groups, graphs and random walks, London Math. Soc. Lecture Note Ser., vol. 436, Cambridge Univ. Press, Cambridge, 2017, pp. 215–249. MR 3644011
- R. Grigorchuk and K. Medynets, Topological full groups are locally embeddable into finite groups, Preprint, arXiv:1105.0719v3, 2012.
- Rostislav I. Grigorchuk and Andrzej Żuk, On a torsion-free weakly branch group defined by a three state automaton, Internat. J. Algebra Comput. 12 (2002), no. 1-2, 223–246. International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000). MR 1902367, DOI 10.1142/S0218196702001000
- M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
- Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
- Branko Grünbaum and G. C. Shephard, Tilings and patterns, W. H. Freeman and Company, New York, 1987. MR 857454
- Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
- F. Hausdorff, Bemerkung über den Inhalt von Punktmengen, Math. Ann. 75 (1914), no. 3, 428–433 (German). MR 1511802, DOI 10.1007/BF01563735
- Richard H. Herman, Ian F. Putnam, and Christian F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827–864. MR 1194074, DOI 10.1142/S0129167X92000382
- Yutaka Ishii, Hyperbolic polynomial diffeomorphisms of $\Bbb C^2$. I. A non-planar map, Adv. Math. 218 (2008), no. 2, 417–464. MR 2407941, DOI 10.1016/j.aim.2007.11.025
- Yutaka Ishii, Hyperbolic polynomial diffeomorphisms of $\Bbb C^2$. II. Hubbard trees, Adv. Math. 220 (2009), no. 4, 985–1022. MR 2483714, DOI 10.1016/j.aim.2008.09.015
- Yutaka Ishii, Hyperbolic polynomial diffeomorphisms of $\Bbb {C}^2$. III: Iterated monodromy groups, Adv. Math. 255 (2014), 242–304. MR 3167483, DOI 10.1016/j.aim.2013.12.031
- K. Juschenko, Non-elementary amenable subgroups of automata groups, arXiv preprint arXiv:1504.00610, 2015.
- Kate Juschenko and Nicolas Monod, Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2) 178 (2013), no. 2, 775–787. MR 3071509, DOI 10.4007/annals.2013.178.2.7
- Kate Juschenko and Tatiana Nagnibeda, Small spectral radius and percolation constants on non-amenable Cayley graphs, Proc. Amer. Math. Soc. 143 (2015), no. 4, 1449–1458. MR 3314060, DOI 10.1090/S0002-9939-2014-12578-0
- Kate Juschenko, Volodymyr Nekrashevych, and Mikael de la Salle, Extensions of amenable groups by recurrent groupoids, Invent. Math. 206 (2016), no. 3, 837–867. MR 3573974, DOI 10.1007/s00222-016-0664-6
- Kate Juschenko and Mikael de la Salle, Invariant means for the wobbling group, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 2, 281–290. MR 3351042
- Kate Juschenko, Nicolás Matte Bon, Nicolas Monod, and Mikael de la Salle, Extensive amenability and an application to interval exchanges, Ergodic Theory Dynam. Systems 38 (2018), no. 1, 195–219. MR 3742543, DOI 10.1017/etds.2016.32
- V. Kaimanovich, Boundary behaviour of Thompson’s group, Preprint.
- V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), no. 3, 457–490. MR 704539
- Martin Kassabov and Igor Pak, Groups of oscillating intermediate growth, Ann. of Math. (2) 177 (2013), no. 3, 1113–1145. MR 3034295, DOI 10.4007/annals.2013.177.3.7
- Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
- A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 81–106 (Russian). MR 0219697
- Michael Keane, Interval exchange transformations, Math. Z. 141 (1975), 25–31. MR 357739, DOI 10.1007/BF01236981
- Harry Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354. MR 109367, DOI 10.1090/S0002-9947-1959-0109367-6
- Y. Lavrenyuk and V. Nekrashevych, On classification of inductive limits of direct products of alternating groups, J. Lond. Math. Soc. (2) 75 (2007), no. 1, 146–162. MR 2302735, DOI 10.1112/jlms/jdl009
- M. Laczkovich, Equidecomposability and discrepancy; a solution of Tarski’s circle-squaring problem, J. Reine Angew. Math. 404 (1990), 77–117. MR 1037431, DOI 10.1515/crll.1990.404.77
- Henri Leon Lebesgue, Leçons sur l’intégration et la recherche des fonctions primitives professées au Collège de France, Cambridge Library Collection, Cambridge University Press, Cambridge, 2009 (French). Reprint of the 1904 original. MR 2857993, DOI 10.1017/CBO9780511701825
- Felix Leinen and Orazio Puglisi, Some results concerning simple locally finite groups of 1-type, J. Algebra 287 (2005), no. 1, 32–51. MR 2134257, DOI 10.1016/j.jalgebra.2004.12.021
- Yash Lodha and Justin Tatch Moore, A nonamenable finitely presented group of piecewise projective homeomorphisms, Groups Geom. Dyn. 10 (2016), no. 1, 177–200. MR 3460335, DOI 10.4171/GGD/347
- Alexander Lubotzky, Group presentation, $p$-adic analytic groups and lattices in $\textrm {SL}_{2}(\textbf {C})$, Ann. of Math. (2) 118 (1983), no. 1, 115–130. MR 707163, DOI 10.2307/2006956
- Russell Lyons and Yuval Peres, Probability on trees and networks, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42, Cambridge University Press, New York, 2016. MR 3616205, DOI 10.1017/9781316672815
- Nicolás Matte Bon, Subshifts with slow complexity and simple groups with the Liouville property, Geom. Funct. Anal. 24 (2014), no. 5, 1637–1659. MR 3261637, DOI 10.1007/s00039-014-0293-4
- Hiroki Matui, Some remarks on topological full groups of Cantor minimal systems, Internat. J. Math. 17 (2006), no. 2, 231–251. MR 2205435, DOI 10.1142/S0129167X06003448
- Konstantin Medynets, Cantor aperiodic systems and Bratteli diagrams, C. R. Math. Acad. Sci. Paris 342 (2006), no. 1, 43–46 (English, with English and French summaries). MR 2193394, DOI 10.1016/j.crma.2005.10.024
- Yu. I. Merzlyakov, Infinite finitely generated periodic groups, Dokl. Akad. Nauk SSSR 268 (1983), no. 4, 803–805 (Russian). MR 693210
- John Milnor, Pasting together Julia sets: a worked out example of mating, Experiment. Math. 13 (2004), no. 1, 55–92. MR 2065568
- J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7. MR 232311
- John Milnor, Growth of finitely generated solvable groups, J. Differential Geometry 2 (1968), 447–449. MR 244899
- Milnor, J., Problem 5603, Amer. Math. Monthly, 75 (1968), 685–686.
- Bojan Mohar, Isoperimetric inequalities, growth, and the spectrum of graphs, Linear Algebra Appl. 103 (1988), 119–131. MR 943998, DOI 10.1016/0024-3795(88)90224-8
- Nicolas Monod, Groups of piecewise projective homeomorphisms, Proc. Natl. Acad. Sci. USA 110 (2013), no. 12, 4524–4527. MR 3047655, DOI 10.1073/pnas.1218426110
- F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275, DOI 10.2307/1968693
- I. Namioka, Følner’s conditions for amenable semi-groups, Math. Scand. 15 (1964), 18–28. MR 180832, DOI 10.7146/math.scand.a-10723
- Volodymyr Nekrashevych, Palindromic subshifts and simple periodic groups of intermediate growth, Ann. of Math. (2) 187 (2018), no. 3, 667–719. MR 3779956, DOI 10.4007/annals.2018.187.3.2
- V. V. Nekrashevych, Self-similar inverse semigroups and groupoids, Ukrainian Mathematics Congress—2001 (Ukrainian), Natsīonal. Akad. Nauk Ukraïni, Īnst. Mat., Kiev, 2002, pp. 176–192. MR 2228863
- Volodymyr Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs, vol. 117, American Mathematical Society, Providence, RI, 2005. MR 2162164, DOI 10.1090/surv/117
- Volodymyr Nekrashevych, Self-similar inverse semigroups and Smale spaces, Internat. J. Algebra Comput. 16 (2006), no. 5, 849–874. MR 2274718, DOI 10.1142/S0218196706003153
- Volodymyr Nekrashevych, A minimal Cantor set in the space of 3-generated groups, Geom. Dedicata 124 (2007), 153–190. MR 2318543, DOI 10.1007/s10711-006-9118-4
- Volodymyr Nekrashevych, Symbolic dynamics and self-similar groups, Holomorphic dynamics and renormalization, Fields Inst. Commun., vol. 53, Amer. Math. Soc., Providence, RI, 2008, pp. 25–73. MR 2477417
- Volodymyr Nekrashevych, Combinatorics of polynomial iterations, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 169–214. MR 2508257, DOI 10.1201/b10617-5
- Volodymyr Nekrashevych, Free subgroups in groups acting on rooted trees, Groups Geom. Dyn. 4 (2010), no. 4, 847–862. MR 2727668, DOI 10.4171/GGD/110
- Peter M. Neumann, Some questions of Edjvet and Pride about infinite groups, Illinois J. Math. 30 (1986), no. 2, 301–316. MR 840129
- J. von Neumann, Zur allgemeinen Theorie des Masses, Fund. Math., vol 13 (1929), 73–116.
- C. St. J. A. Nash-Williams, Random walk and electric currents in networks, Proc. Cambridge Philos. Soc. 55 (1959), 181–194. MR 124932, DOI 10.1017/s0305004100033879
- Ricardo Antonio Oliva, On the combinatorics of external rays in the dynamics of the complex Henon map, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–Cornell University. MR 2697417
- A. Ju. Ol′šanskiĭ, On the question of the existence of an invariant mean on a group, Uspekhi Mat. Nauk 35 (1980), no. 4(214), 199–200 (Russian). MR 586204
- Alexander Yu. Ol′shanskii and Mark V. Sapir, Non-amenable finitely presented torsion-by-cyclic groups, Publ. Math. Inst. Hautes Études Sci. 96 (2002), 43–169 (2003). MR 1985031
- D. V. Osin, Elementary classes of groups, Mat. Zametki 72 (2002), no. 1, 84–93 (Russian, with Russian summary); English transl., Math. Notes 72 (2002), no. 1-2, 75–82. MR 1942584, DOI 10.1023/A:1019869105364
- Denis V. Osin, $L^2$-Betti numbers and non-unitarizable groups without free subgroups, Int. Math. Res. Not. IMRN 22 (2009), 4220–4231. MR 2552302, DOI 10.1093/imrn/rnp085
- Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
- Jean-Paul Pier, Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 767264
- A. Rejali and A. Yousofzadeh, Configuration of groups and paradoxical decompositions, Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 1, 157–172. MR 2809910
- Joseph Max Rosenblatt, A generalization of Følner’s condition, Math. Scand. 33 (1973), 153–170. MR 333068, DOI 10.7146/math.scand.a-11481
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- Volker Runde, Lectures on amenability, Lecture Notes in Mathematics, vol. 1774, Springer-Verlag, Berlin, 2002. MR 1874893, DOI 10.1007/b82937
- Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
- Jan-Christoph Schlage-Puchta, A $p$-group with positive rank gradient, J. Group Theory 15 (2012), no. 2, 261–270. MR 2900227, DOI 10.1515/jgt.2011.101
- Dan Segal, The finite images of finitely generated groups, Proc. London Math. Soc. (3) 82 (2001), no. 3, 597–613. MR 1816690, DOI 10.1112/plms/82.3.597
- Said Sidki, Automorphisms of one-rooted trees: growth, circuit structure, and acyclicity, J. Math. Sci. (New York) 100 (2000), no. 1, 1925–1943. Algebra, 12. MR 1774362, DOI 10.1007/BF02677504
- Said Sidki, Finite automata of polynomial growth do not generate a free group, Geom. Dedicata 108 (2004), 193–204. MR 2112674, DOI 10.1007/s10711-004-2368-0
- Otto Schreier, Die Untergruppen der freien Gruppen, Abh. Math. Sem. Univ. Hamburg 5 (1927), no. 1, 161–183 (German). MR 3069472, DOI 10.1007/BF02952517
- V. Sushchansky, Periodic permutation $p$-groups and the unrestricted Burnside problem, DAN SSSR., 247(3):557–562, 1979 (Russian).
- S. Świerczkowski, On a free group of rotations of the Euclidean space, Nederl. Akad. Wetensch. Proc. Ser. A 61 = Indag. Math. 20 (1958), 376–378. MR 0096732
- A. Tarski, Algebraische Fassung de Massproblems, Fund. Math. 31 (1938), 47–66.
- Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
- M. Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125, Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6. MR 1943006, DOI 10.1007/978-3-662-10451-4
- M. Takesaki, Theory of operator algebras. III, Encyclopaedia of Mathematical Sciences, vol. 127, Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 8. MR 1943007, DOI 10.1007/978-3-662-10453-8
- Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
- Marcelo Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut. 19 (2006), no. 1, 7–100. MR 2219821, DOI 10.5209/rev_{R}EMA.2006.v19.n1.16621
- G. Vitali, Sul problema della misura dei gruppi di punti di una retta, Bologna, Tip. Camberini e Parmeggiani (1905).
- Stan Wagon, The Banach-Tarski paradox, Cambridge University Press, Cambridge, 1993. With a foreword by Jan Mycielski; Corrected reprint of the 1985 original. MR 1251963
- Wolfgang Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000. MR 1743100, DOI 10.1017/CBO9780511470967
- Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geometry 2 (1968), 421–446. MR 248688
- Adam Woryna, The rank and generating set for iterated wreath products of cyclic groups, Comm. Algebra 39 (2011), no. 7, 2622–2631. MR 2821737, DOI 10.1080/00927872.2010.544697