## Duke Mathematical Journal

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

Mikhail Ershov

#### 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

#### Article information

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

Dates
First available in Project Euclid: 20 October 2008

https://projecteuclid.org/euclid.dmj/1224508839

Digital Object Identifier
doi:10.1215/00127094-2008-053

Mathematical Reviews number (MathSciNet)
MR2449949

Zentralblatt MATH identifier
1162.20018

#### Citation

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

#### 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.
• 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.