Duke Mathematical Journal

Golod-Shafarevich groups with property ($T$) and Kac-Moody groups

Mikhail Ershov

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


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

Article information

Duke Math. J. Volume 145, Number 2 (2008), 309-339.

First available: 20 October 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E18: Limits, profinite groups
Secondary: 20F05: Generators, relations, and presentations 20E42: Groups with a $BN$-pair; buildings [See also 51E24]


Ershov, Mikhail. Golod-Shafarevich groups with property ( T ) and Kac-Moody groups. Duke Mathematical Journal 145 (2008), no. 2, 309--339. doi:10.1215/00127094-2008-053. http://projecteuclid.org/euclid.dmj/1224508839.

Export citation


  • 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.
  • 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.
  • B. Bekka, P. De La Harpe, and A. Valette, Kazhdan's Property $(T)$, New Math. Monogr. 11, Cambridge Univ. Press, Cambridge, 2008.
  • P.-E. Caprace, On $2$-spherical Kac-Moody groups and their central extensions, Forum Math. 19 (2007), 763--781.
  • 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.
  • L. Carbone and H. Garland, Existence of lattices in Kac-Moody groups over finite fields, Commun. Contemp. Math. 5 (2003), 813--867.
  • 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.
  • —, ``Measures finiment additives et paradoxes'' in Autour du centenaire Lebesgue (Lyon, 2001), Panor. Syntheses 18, Soc. Math. France, Montrouge, 2004, 39--61.
  • A. Devillers and B. MüHlherr, On the simple connectedness of certain subsets of buildings, Forum Math. 19 (2007), 955--970.
  • J. Dymara and T. Januszkiewicz, Cohomology of buildings and their automorphism groups, Invent. Math. 150 (2002), 579--627.
  • M. Ershov, Finite presentability of $\rm SL_1(D)$, Israel J. Math. 158 (2007), 297--347.
  • E. S. Golod, On nil-algebras and finitely approximable p-groups (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 273--276.
  • 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.
  • J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer, New York, 1972.
  • 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.
  • 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.
  • M. Kassabov, Universal lattices and unbounded rank expanders, Invent. Math. 170 (2007), 297--326.
  • M. Kassabov and N. Nikolov, Universal lattices and property tau, Invent. Math. 165 (2006), 209--224.
  • 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.
  • H. Koch, Galois Theory of $p$-Extensions, Springer Monogr. Math., Springer, Berlin, 2002.
  • 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]
  • —, New lower bounds on subgroup growth and homology growth, preprint, to appear in Proc. Lond. Math. Soc. (3), preprint,\arxivmath/0512261v3[math.GR]
  • 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.
  • A. Lubotzky, Group presentation, $p$-adic analytic groups and lattices in $\rm SL\sb2(\bf C)$, Ann. of Math. (2) 118 (1983), 115--130.
  • —, Discrete Groups, Expanding Graphs and Invariant Measures, Progr. Math. 125, Birkhäuser, Basel, 1994.
  • 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.
  • J. Morita, Commutator relations in Kac-Moody groups, Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), 21--22.
  • B. RéMy, Construction de réseaux en théorie de Kac-Moody, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 475--478.
  • —, Groupes de Kac-Moody déployés et presque déployés, Astérisque 277, Soc. Math. France, Montrouge, 2002.
  • —, Topological simplicity, commensurator super-rigidity and non-linearities of Kac-Moody groups, Geom. Funct. Anal. 14 (2004), 810--852.
  • B. RéMy and M. Ronan, Topological groups of Kac-Moody type, right-angled twinnings and their lattices, Comment. Math. Helv. 81 (2006), 191--219.
  • Y. Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), 1--54.
  • R. Steinberg, Lectures on Chevalley Groups, notes prepared by J. Faulkner and R. Wilson, Yale Univ., New Haven, Conn., 1968.
  • J. Tits, Ensembles ordonnés, immeubles et sommes amalgamées, Bull. Soc. Math. Belg. Sér. A 38 (1986), 367--387.
  • —, Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra 105 (1987), 542--573.
  • A. Vershik, ``Amenability and approximation of infinite groups'' in Selected Translations, Selecta Math. Soviet. 2, no. 4, Birkhäuser, Secaucus, N. J., 1982, 311--330.
  • E. Zelmanov, ``On groups satisfying the Golod-Shafarevich condition'' in New Horizons in Pro-$p$ Groups, Progr. Math. 184, Birkhäuser, Boston, 2000, 223--232.
  • —, ``Infinite algebras and pro-$p$ groups'' in Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Progr. Math. 248, Birkhäuser, Basel, 2005, 403--413.