Golod-Shafarevich groups with property ($T$) and Kac-Moody groups
Mikhail Ershov
Source: Duke Math. J. Volume 145, Number 2
(2008), 309-339.
Abstract
We construct Golod-Shafarevich groups with property $(T)$ and thus provide counterexamples to a conjecture stated in a recent article of Zelmanov [Z2]. Explicit examples of such groups are given by lattices in certain topological Kac-Moody groups over finite fields. We provide several applications of this result, including examples of residually finite torsion nonamenable groups
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1224508839
Digital Object Identifier: doi:10.1215/00127094-2008-053
Mathematical Reviews number (MathSciNet): MR2449949
Zentralblatt MATH identifier: 1162.20018
References
P. Abramenko, ``Finiteness properties of groups acting on twin buildings'' in Groups: Topological, Combinatorial and Arithmetic Aspects, London Math. Soc. Lecture Note Ser. 311, Cambridge Univ. Press, Cambridge, 2004, 21--26.
Mathematical Reviews (MathSciNet): MR2073344
Zentralblatt MATH: 1135.20304
P. Abramenko and B. MüHlherr, Présentations de certaines $BN$-paires jumelées comme sommes amalgamées, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), 701--706.
Mathematical Reviews (MathSciNet): MR1483702
Digital Object Identifier: doi:10.1016/S0764-4442(97)80044-4
B. Bekka, P. De La Harpe, and A. Valette, Kazhdan's Property $(T)$, New Math. Monogr. 11, Cambridge Univ. Press, Cambridge, 2008.
Mathematical Reviews (MathSciNet): MR2415834
P.-E. Caprace, On $2$-spherical Kac-Moody groups and their central extensions, Forum Math. 19 (2007), 763--781.
Mathematical Reviews (MathSciNet): MR2350773
Digital Object Identifier: doi:10.1515/FORUM.2007.031
Zentralblatt MATH: 1140.20028
L. Carbone, M. Ershov, and G. Ritter, Abstract simplicity of complete Kac-Moody groups over finite fields, J. Pure Appl. Algebra 212 (2008), 2147--2162.
Mathematical Reviews (MathSciNet): MR2418160
Digital Object Identifier: doi:10.1016/j.jpaa.2008.03.023
Zentralblatt MATH: 1157.20018
L. Carbone and H. Garland, Existence of lattices in Kac-Moody groups over finite fields, Commun. Contemp. Math. 5 (2003), 813--867.
Mathematical Reviews (MathSciNet): MR2017720
Digital Object Identifier: doi:10.1142/S0219199703001117
Zentralblatt MATH: 1070.17010
P. De La Harpe, ``Uniform growth in groups of exponential growth'' in Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), Geom. Dedicata 95, Springer, Dordrecht, Netherlands, 2002, 1--17.
Mathematical Reviews (MathSciNet): MR1950882
Digital Object Identifier: doi:10.1023/A:1021273024728
Zentralblatt MATH: 1025.20027
—, ``Measures finiment additives et paradoxes'' in Autour du centenaire Lebesgue (Lyon, 2001), Panor. Syntheses 18, Soc. Math. France, Montrouge, 2004, 39--61.
Mathematical Reviews (MathSciNet): MR2143416
Zentralblatt MATH: 1084.43001
A. Devillers and B. MüHlherr, On the simple connectedness of certain subsets of buildings, Forum Math. 19 (2007), 955--970.
Mathematical Reviews (MathSciNet): MR2367950
Digital Object Identifier: doi:10.1515/FORUM.2007.037
Zentralblatt MATH: 05237810
J. Dymara and T. Januszkiewicz, Cohomology of buildings and their automorphism groups, Invent. Math. 150 (2002), 579--627.
Mathematical Reviews (MathSciNet): MR1946553
Digital Object Identifier: doi:10.1007/s00222-002-0242-y
Zentralblatt MATH: 1140.20308
M. Ershov, Finite presentability of $\rm SL_1(D)$, Israel J. Math. 158 (2007), 297--347.
Mathematical Reviews (MathSciNet): MR2342469
Digital Object Identifier: doi:10.1007/s11856-007-0015-9
Zentralblatt MATH: 1125.20034
E. S. Golod, On nil-algebras and finitely approximable p-groups (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 273--276.
Mathematical Reviews (MathSciNet): MR0161878
E. S. Golod and I. R. šAfarevič [Shafarevich], On the class field tower (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261--272.
Mathematical Reviews (MathSciNet): MR0161852
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer, New York, 1972.
Mathematical Reviews (MathSciNet): MR0323842
A. Jaikin, M. Kassabov, and N. Nikolov, Property $(\tau)$ and subgroup growth, in preparation.
V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990.
Mathematical Reviews (MathSciNet): MR1104219
V. G. Kac and D. H. Peterson, ``Defining relations of certain infinite-dimensional groups'' in The Mathematical Heritage of Élie Cartan (Lyon, 1984), Astérisque 1985, Numero Hors Serie, Soc. Math. France, Montrouge, 165--208.
Mathematical Reviews (MathSciNet): MR0837201
Zentralblatt MATH: 0625.22014
M. Kassabov, Universal lattices and unbounded rank expanders, Invent. Math. 170 (2007), 297--326.
Mathematical Reviews (MathSciNet): MR2342638
Digital Object Identifier: doi:10.1007/s00222-007-0064-z
Zentralblatt MATH: 1140.20039
M. Kassabov and N. Nikolov, Universal lattices and property tau, Invent. Math. 165 (2006), 209--224.
Mathematical Reviews (MathSciNet): MR2221141
Digital Object Identifier: doi:10.1007/s00222-005-0498-0
Zentralblatt MATH: 1139.19003
D. A. KažDan [Kazhdan], On the connection of the dual space of a group with the structure of its closed subgroups (in Russian), Funkcional. Anal. i Priložen. 1 (1967), 71--74.
Mathematical Reviews (MathSciNet): MR0209390
H. Koch, Galois Theory of $p$-Extensions, Springer Monogr. Math., Springer, Berlin, 2002.
Mathematical Reviews (MathSciNet): MR1930372
Zentralblatt MATH: 1023.11002
M. Lackenby, Large groups, property ($\tau$) and the homology growth of subgroups, to appear in Math. Proc. Cambridge Philos. Soc., preprint,\arxivmath/0509036v3[math.GR]
Mathematical Reviews (MathSciNet): MR2496348
Zentralblatt MATH: 05555730
Digital Object Identifier: doi:10.1017/S0305004108002089
—, New lower bounds on subgroup growth and homology growth, preprint, to appear in Proc. Lond. Math. Soc. (3), preprint,\arxivmath/0512261v3[math.GR]
Mathematical Reviews (MathSciNet): MR2481949
Zentralblatt MATH: 05530817
Digital Object Identifier: doi:10.1112/plms/pdn032
M. Lackenby, D. D. Long, and A. W. Reid, Covering spaces of arithmetic $3$-orbifolds, Int. Math. Res. Not. IMRN 2008, no. 12, art. ID rnn036.
Mathematical Reviews (MathSciNet): MR2426753
Zentralblatt MATH: 1155.57022
A. Lubotzky, Group presentation, $p$-adic analytic groups and lattices in $\rm SL\sb2(\bf C)$, Ann. of Math. (2) 118 (1983), 115--130.
Mathematical Reviews (MathSciNet): MR0707163
Digital Object Identifier: doi:10.2307/2006956
JSTOR: links.jstor.org
—, Discrete Groups, Expanding Graphs and Invariant Measures, Progr. Math. 125, Birkhäuser, Basel, 1994.
Mathematical Reviews (MathSciNet): MR1308046
Zentralblatt MATH: 0826.22012
A. Lubotzky and A. $\dot\rm z$uk, On Property ($\tau$), book in preparation.
G. A. Margulis, Explicit constructions of expanders (in Russian), Problemy Peredači Informacii 9, no. 4 (1973), 71--80.; English translation in Problems of Information Transmission 9 (1973), 325--332.
Mathematical Reviews (MathSciNet): MR0484767
J. Morita, Commutator relations in Kac-Moody groups, Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), 21--22.
Mathematical Reviews (MathSciNet): MR0892949
Digital Object Identifier: doi:10.3792/pjaa.63.21
Project Euclid: euclid.pja/1195514022
B. RéMy, Construction de réseaux en théorie de Kac-Moody, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 475--478.
Mathematical Reviews (MathSciNet): MR1715140
Digital Object Identifier: doi:10.1016/S0764-4442(00)80044-0
—, Groupes de Kac-Moody déployés et presque déployés, Astérisque 277, Soc. Math. France, Montrouge, 2002.
Mathematical Reviews (MathSciNet): MR1909671
—, Topological simplicity, commensurator super-rigidity and non-linearities of Kac-Moody groups, Geom. Funct. Anal. 14 (2004), 810--852.
Mathematical Reviews (MathSciNet): MR2084981
B. RéMy and M. Ronan, Topological groups of Kac-Moody type, right-angled twinnings and their lattices, Comment. Math. Helv. 81 (2006), 191--219.
Mathematical Reviews (MathSciNet): MR2208804
Y. Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), 1--54.
Mathematical Reviews (MathSciNet): MR1767270
Digital Object Identifier: doi:10.1007/s002220000064
Zentralblatt MATH: 0978.22010
R. Steinberg, Lectures on Chevalley Groups, notes prepared by J. Faulkner and R. Wilson, Yale Univ., New Haven, Conn., 1968.
Mathematical Reviews (MathSciNet): MR0466335
Zentralblatt MATH: 0307.22001
J. Tits, Ensembles ordonnés, immeubles et sommes amalgamées, Bull. Soc. Math. Belg. Sér. A 38 (1986), 367--387.
Mathematical Reviews (MathSciNet): MR0885545
—, Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra 105 (1987), 542--573.
Mathematical Reviews (MathSciNet): MR0873684
Digital Object Identifier: doi:10.1016/0021-8693(87)90214-6
Zentralblatt MATH: 0626.22013
A. Vershik, ``Amenability and approximation of infinite groups'' in Selected Translations, Selecta Math. Soviet. 2, no. 4, Birkhäuser, Secaucus, N. J., 1982, 311--330.
Mathematical Reviews (MathSciNet): MR0721030
Zentralblatt MATH: 0533.22007
E. Zelmanov, ``On groups satisfying the Golod-Shafarevich condition'' in New Horizons in Pro-$p$ Groups, Progr. Math. 184, Birkhäuser, Boston, 2000, 223--232.
Mathematical Reviews (MathSciNet): MR1765122
Zentralblatt MATH: 0974.20022
—, ``Infinite algebras and pro-$p$ groups'' in Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Progr. Math. 248, Birkhäuser, Basel, 2005, 403--413.
Mathematical Reviews (MathSciNet): MR2195460
Digital Object Identifier: doi:10.1007/3-7643-7447-0_11
Zentralblatt MATH: 1115.20023
Duke Mathematical Journal
