Lie algebras and growth in branch groups

Pacific Journal of Mathematics

Volumn 218, Issue 2, 2005, Pages 241-282

  Bartholdi, Laurent ^a  

^a EPFL   (Switzerland)

Author keywords

Growth of groups; Lie algebra; Lower central series

Indexed keywords

EID: 18844410779     PISSN: 00308730     EISSN: None     Source Type: Journal    
DOI: 10.2140/pjm.2005.218.241     Document Type: Article

References (56)

1. [Bartholdi 2000] L. Bartholdi, Croissance des groupes agissant sur des arbres, Ph.D. Thesis, Université de Genève, 2000.
2. [Bartholdi ≥ 2005] L. Bartholdi, "Self-similar Lie algebras and groups", in preparation.
3. [Bartholdi and Grigorchuk 2000a] L. Bartholdi and R. I. Grigorchuk, "Lie methods in growth of groups and groups of finite width", pp. 1-27 in Computational and geometric aspects of modem algebra (Edinburgh, 1998), edited by M. Atkinson et al., London Math. Soc. Lecture Note Ser. 275, Cambridge Univ. Press, Cambridge, 2000. MR 2001h:20046 Zbl 1032.20026
4. [Bartholdi and Grigorchuk 2000b] L. Bartholdi and R. I. Grigorchuk, "On the spectrum of Hecke type operators related to some fractal groups", pp. 5-45 in Din. Sist., Avtom. i Beskon. Gruppy, Tr. Mat. Inst. Steklova 231, 2000. MR 2002d:37017 Zbl 01729296
5. [Bartholdi and Grigorchuk 2002] L. Bartholdi and R. I. Grigorchuk, "On parabolic subgroups and Hecke algebras of some fractal groups", Serdica Math. J. 28:1 (2002), 47-90. MR 2003c:20027 Zbl 1011.20028
6. [Bartholdi and Sidki 2003] L. Bartholdi and S. N. Sidki, "The automorphism tower of groups acting on rooted trees", preprint, 2003. To appear in Trans. Amer. Math. Soc. math.GR/0308127
7. [Bergman 1978] G. M. Bergman, "A note on growth functions of algebras and semigroups", 1978.
8. [Blessenohl and Laue 1993] D. Blessenohl and H. Laue, "A basis construction for free Lie algebras", Exposition. Math. 11:2(1993), 145-152. MR 94d: 17006 Zbl 0782.17005
9. [Camina 2000] R. Camina, "The Nottingham group", pp. 205-221 in New horizons in pro-p groups, Progr. Math. 184, Birkhäuser, Boston, 2000. MR2001f:20054 Zbl 0977.20020
10. [Garanti 1997] A. Garanti, "Presenting the graded Lie algebra associated to the Nottingham group", J. Algebra 198:1 (1997), 266-289. MR99b:17019 Zbl 0977.17016
11. [Ceccherini-Silberstein et al. 2001] T. Ceccherini-Silberstein, F. Scarabotti, and F. Tolli, "The top of the lattice of normal subgroups of the Grigorchuk group", J. Algebra 246:1 (2001), 292-310. MR 2002k:20048 Zbl 0998.20036
12. [Fabrykowski and Gupta 1991] J. Fabrykowski and N. Gupta, "On groups with sub-exponential growth functions, II", J. Indian Math. Society (N.S.) 56:1-4 (1991), 217-228. MR 93g:20053 Zbl 0868.20029
13. [GAP 2002] The GAP Group, "GAP: Groups, Algorithms, and Programming", 2002, available at http://www.gap-system.org/. Version 4.3.
14. [Gelfand and Kirillov 1966] I. M. Gel'fand and A. A. Kirillov, "On fields connected with the enveloping algebras of Lie algebras", Dokl. Akad. Nauk SSSR 167 (1966), 503-505. In Russian;
15. translated in Sov. Math. Dokl. 7 (1966), 407-409. MR 33 #4108 Zbl 0149.02903
16. [Grigorchuk 1980] R. I. Grigorčuk, "On Burnside's problem on periodic groups", Funktsional. Anal. i Prilozhen. 14:1 (1980), 53-54. In Russian;
17. translated in Functional Anal. Appl. 14:1 (1980), 4143. MR 81m:20045 Zbl 0595.20029
18. [Grigorchuk 1983] R. I. Grigorčuk, "On the Milnor problem of group growth", Dokl. Akad. Nauk SSSR 271:1 (1983), 30-33. In Russian;
19. translated in Soviet Math. Doklady 28:1 (1983), 23-26. MR 85g:20042 Zbl 0547.20025
20. [Grigorchuk 1989] R. I. Grigorčuk, "On the Hilbert-Poincaré series of graded algebras that are associated with groups", Mat. Sb. 180:2 (1989), 207-225, 304. In Russian;
21. translated in Math. USSR Sb. 66:1 (1990), 211-229. MR 90j:20063 Zbl 0695.16009
22. [Grigorchuk 2000] R. I. Grigorchuk, "Just infinite branch groups", pp. 121-179 in New horizons in pro-p groups, edited by D. S. Marcus du Sautoy and A. Shalev, Progr. Math. 184, Birkhäuser, Boston, 2000. MR 2002f:20044 Zbl 0982.20024
23. [Gupta and Sidki 1983] N. Gupta and S. Sidki, "On the Burnside problem for periodic groups", Math. Z. 182:3 (1983), 385-388. MR 84g:20075 Zbl 0513.20024
24. [Hall 1950] M. Hall, Jr., "A basis for free Lie rings and higher commutators in free groups", Proc. Amer. Math. Soc. 1 (1950), 575-581. MR 12,388a Zbl 0039.26302
25. [de la Harpe 2000] P. de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 2001i:20081 Zbl 0965.20025
26. [Huppert and Blackburn 1982] B. Huppert and N. Blackburn, Finite groups. II, vol. 242, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1982. AMD, 44. MR84i:20001a Zbl 0477.20001
27. [Jacobson 1941] N. Jacobson, "Restricted Lie algebras of characteristic p", Trans. Amer. Math. Soc. 50 (1941), 15-25. MR 3,103g JFM 0025.30301
28. [Jennings 1941] S. A. Jennings, "The structure of the group ring of a p-group over a modular field", Trans. Amer. Math. Soc. 50 (1941), 175-185. MR 3,34f JFM 0025.24401
29. [Jennings 1954] S. A. Jennings, "Substitution groups of formal power series", Canadian J. Math. 6 (1954), 325-340. MR 15,853f Zbl 0058.02201
30. [Kaloujnine 1948] L. Kaloujnine, "La structure des p-groupes de Sylow des groupes symétriques finis", Ann. Sci. École Norm. Sup. (3) 65 (1948), 239-276. MR 10,505b Zbl 0034.30501
31. [Klaas et al. 1997] G. Klaas, C. R. Leedham-Green, and W. Plesken, Linear pro- p-groups of finite width, vol. 1674, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1997. MR 98m:20028 Zbl 0901.20013
32. [Krause and Lenagan 1985] G. R. Krause and T. H. Lenagan, Growth of algebras and Gel'fand-Kirillov dimension, Research Notes in Mathematics 116, Pitman, Boston, 1985. Revised edition,
33. Grad. Stud. Math. 22, Amer. Math. Soc., Providence, 2000. MR 86g:16001 Zbl 0564.16001
34. [Kukin 1978] G. P. Kukin, "Bases of a free Lie algebra", Mat. Zametki 24:3 (1978), 375-382, 447. MR 80b: 17003 Zbl 0404.17013
35. [Lazard 1954] M. Lazard, "Sur les groupes nilpotents et les anneaux de Lie", Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 101-190. MR 19,529b Zbl 0055.25103
36. [Leedham-Green 1994] C. R. Leedham-Green, "Pro-p-groups of finite coclass", J. London Math. Soc. (2) 50:1 (1994), 43-48. MR 95j:20021 Zbl 0817.20031
37. [Li 1996] W. C. W. Li, Number theory with applications, vol. 7, Series on University Mathematics, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. MR 98b:11001 Zbl 0849.11006
38. [Lothaire 1990] M. Lothaire, Mots, Langue, Raisonnement, Calcul. [Language, Reasoning, Computation], Hermès, Paris, 1990. MR 94g:05004 Zbl 0862.05001
39. [Magnus 1940] W. Magnus, "Über Gruppen und zugeordnete Liesche Ringe", J. Reine Angew. Math. 182 (1940), 142-149. MR2,214d Zbl 0025.24201 JFM 66.1208.03
40. [Müller 1996] T. Müller, "Subgroup growth of free products", Invent. Math. 126:1 (1996), 111-131. MR 97f :20031 Zbl 0862.20019
41. [Passi 1979] I. B. S. Passi, Group rings and their augmentation ideals, vol. 715, Lecture Notes in Mathematics, Springer, Berlin, 1979. MR 80k:20009 Zbl 0405.20007
42. [Passi and Sehgal 1975] I. B. S. Passi and S. K. Sehgal, "Lie dimension subgroups", Comm. Algebra 3 (1975), 59-73. MR 51 #5725 Zbl 0319.20055
43. [Petrogradsky 1997] V. M. Petrogradsky, "On Lie algebras with nonintegral q-dimensions", Proc. Amer. Math. Soc. 125:3 (1997), 649-656. MR 97e:16055 Zbl 0873.17014
44. [Petrogradsky 1999] V. M. Petrogradsky, "Growth of finitely generated polynilpotent Lie algebras and groups, generalized partitions, and functions analytic in the unit circle", Internat. J. Algebra Comput. 9:2 (1999), 179-212. MR 2000i:17007 Zbl 0931.17007
45. [Quillen 1968] D. G. Quillen, "On the associated graded ring of a group ring", J. Algebra 10 (1968), 411-418. MR 38 #245 Zbl 0192.35803
46. [Reutenauer 1993] C. Reutenauer, Free Lie algebras, vol. 7, London Mathematical Society Monographs. New Series, The Clarendon Press Oxford University Press, New York, 1993. Oxford Science Publications. MR 94j:17002 Zbl 0798.17001
47. [Rozhkov 1996] A. V. Rozhkov, "Lower central series of a group of tree automorphisms", Mat. Zametki 60:2 (1996), 225-237, 319. MR 98b:20048 Zbl 0899.20014
48. [Shalev 1995a] A. Shalev, "Some problems and results in the theory of pro-p groups", pp. 528-542 in Groups '93 Galway/St. Andrews (Galway, 1993), vol. 2, edited by C. M. Campbell et al., London Math. Soc. Lecture Note Ser. 212, Cambridge Univ. Press, Cambridge, 1995. MR 96k:20040 Zbl 0866.20020
49. [Shalev 1995b] A. Shalev, "Finite p-groups", pp. 401-450 in Finite and locally finite groups (Istanbul, 1994), edited by A. V. B. B. Hartley, G. M. Seitz and R. M. Bryant, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 471, Kluwer Acad. Publ., Dordrecht, 1995. MR 97h:20023 Zbl 0839.20028
50. [Shalev and Zelmanov 1992] A. Shalev and E. I. Zel'manov, "Pro-p groups of finite coclass", Math. Proc. Cambridge Philos. Soc. 111:3 (1992), 417-421. MR 93e:20030 Zbl 0813.20030
51. [Sidki 1987] S. Sidki, "On a 2-generated infinite 3-group: subgroups and automorphisms", J. Algebra 110:1 (1987), 24-55. MR 89b:20081b Zbl 0623.20025
52. [Širšov 1962] A. I. Širšov, "On the bases of a free Lie algebra", Algebra i Logika Sem. 1:1 (1962), 14-19. MR 27 #183
53. [Strade and Farnsteiner 1988] H. Strade and R. Farnsteiner, Modular Lie algebras and their representations, Pure Appl. Math. 116, Dekker, New York, 1988. MR 89h:17021 Zbl 0648.17003
54. [Vieira 1998] A. C. Vieira, "On the lower central series and the derived series of the Gupta-Sidki 3-group", Comm. Algebra 26:4 (1998), 1319-1333. MR 99i:20041 Zbl 0899.20015
55. [Vieira 1999] A. C. Vieira, "On some series of normal subgroups of the Gupta-Sidki 3-group", pp. 728-737 in Groups St. Andrews 1997 (Bath, 1997), vol. 2, edited by C. M. Campbell et al., London Math. Soc. Lect. Note Ser. 261, Cambridge Univ. Press, 1999. MR 2000h:20076 Zbl 1004.20502
56. [Zassenhaus 1940] H. Zassenhaus, "Ein Verfahren, jeder endlichen p-Gruppe einen Lie-Ring mit der Charakteristik p zuzuordnen", Abhandlungen Math. Sem. Univ. Hamburg 13 (1940), 200-207. Zbl 0021.20001 JFM 65.0091.01