COCOMPACT LATTICES ON Ãn BUILDINGS

CAPDEBOSCQ, INNA ; RUMYNIN, DMITRIY ; THOMAS, ANNE

Glasgow mathematical journal, Tome 57 (2015) no. 2, pp. 241-262

Résumé

We construct cocompact lattices Γ'0 < Γ0 in the group G = PGLd$({\mathbb{F}_q(\!(t)\!)\!})$ which are type-preserving and act transitively on the set of vertices of each type in the building Δ associated to G. These lattices are commensurable with the lattices of Cartwright–Steger Isr. J. Math.103 (1998), 125–140. The stabiliser of each vertex in Γ'0 is a Singer cycle and the stabiliser of each vertex in Γ0 is isomorphic to the normaliser of a Singer cycle in PGLd(q). We show that the intersections of Γ'0 and Γ0 with PSLd$({\mathbb{F}_q(\!(t)\!)\!})$ are lattices in PSLd$({\mathbb{F}_q(\!(t)\!)\!})$, and identify the pairs (d, q) such that the entire lattice Γ'0 or Γ0 is contained in PSLd$({\mathbb{F}_q(\!(t)\!)\!})$. Finally we discuss minimality of covolumes of cocompact lattices in SL3$({\mathbb{F}_q(\!(t)\!)\!})$. Our proofs combine the construction of Cartwright–Steger Isr. J. Math.103 (1998), 125–140 with results about Singer cycles and their normalisers, and geometric arguments.

Détail

DOI : 10.1017/S0017089514000287

Bibliographie

[1] 1.Bass, H., Covering theory for graphs of groups J. Pure Appl. Algebra 89 (1993), 3–47. Google Scholar

[2] 2.Bass, H. and Lubotzky, A.. Tree lattices, in Progress in Mathematics, vol. 176 (Bass, H., Carbone, L., Lubotzky, A., Rosenberg, G. and Tits, J., Editors) (Birkhäuser Boston, Inc., Boston, MA, 2001), xiv+233. Google Scholar

[3] 3.Borel, A. and Harder, G., Existence of discrete cocompact subgroups of reductive groups over local fields J. Reine Angew. Math. 298 (1978), 53–64. Google Scholar

[4] 4.Borel, A. and Chandra, H., Arithmetic subgroups of algebraic groups Ann. Math. 75 (1962), 485–535. Google Scholar | DOI

[5] 5.Bridson, M. R. and Haefliger, A., Metric spaces of non-positive curvature (Springer-Verlag, Berlin, 1999). Google Scholar | DOI

[6] 6.Cartwright, D. I., Mantero, A. M., Steger, T. and Zappa, A., Groups acting simply transitively on the vertices of a building of type à I, Geom. Dedicata 47 (1993), 143–166. Google Scholar

[7] 7.Cartwright, D. I. and Steger, T., A family of Ögroups Isr. J. Math. 103 (1998), 125–140. Google Scholar

[8] 8.Caprace, P.-E. and Rémy, B., Simplicity and superrigidity of twin building lattices, Invent. Math. 176 (2009), 169–221. Google Scholar | DOI

[9] 9.Capdeboscq, I. K. and Thomas, A., Lattices in complete rank 2 Kac–Moody groups J. Pure Appl. Algebr. 216 (2011), 1348–1371. Google Scholar | DOI

[10] 10.Cossidente, A. and De Resmini, M. J., Remarks on singer cyclic groups and their normalizers, Des. Codes Cryptogr. 32 (2004), 97–102. Google Scholar

[11] 11.Essert, J., A geometric construction of panel-regular lattices for buildings of type à and Algebr. Geom. Topol. 13 (2013), 1531–1578. Google Scholar | DOI

[12] 12.Figá-Talamanca, A. and Nebbia, C., Harmonic analysis and representation theory for groups acting on homogeneous trees, London Mathematical Society Lecture Note Series, vol. 162 (Cambridge University Press, 1991). Google Scholar | DOI

[13] 13.Gelfand, I. M., Graev, M. I. and Piatetsky-Shapiro, I., Representation theory and automorphic functions (Saunders, San Francisco, 1969). Google Scholar

[14] 14.Gorenstein, D., Lyons, R. and Solomon, R., The classification of finite simple groups, Number 3, Part I. Mathematical Surveys and Monographs, vol. 40 (American Mathematical Society, Providence, RI, 1998). Google Scholar

[15] 15.Jacobson, N., PI–algebras. An introduction, Lecture Notes in Mathematics, vol. 441 (Springer-Verlag, Berlin-New York, 1975). Google Scholar

[16] 16.Kantor, W. M., Liebler, R. A. and Tits, J., On discrete chamber-transitive automorphism groups of affine buildings, Bull. Am. Math. Soc. 16 (1987), 129–133. Google Scholar | DOI

[17] 17.Kazhdan, D. A. and Margulis, G. A., A proof of Selberg's hypothesis, Mat. Sb. (N.S.) 75 (117) (1968), 163–168. Google Scholar

[18] 18.Lubotzky, A., Lattices of minimal covolume in SL : a nonarchimedean analogue of Siegel's theorem μ≥π/21 J. Am. Math. Soc. 3 (1990), 961–975. Google Scholar

[19] 19.Lubotzky, A., Lattices in rank one Lie groups over local fields Geom. Funct. Anal. 1 (1991), 406–431. Google Scholar | DOI

[20] 20.Lubotzky, A. and Weigel, Th., Lattices of minimal covolume in SL over local fields, Proc. London Math. Soc. 78 (1999), 283–333. Google Scholar

[21] 21.Margulis, G., Discrete subgroups of semisimple Lie froups, (Springer-Verlag, Berlin, 1991). Google Scholar

[22] 22.Mostow, G. D. and Tamagawa, T., On the cocompactness of arithmetically defined homogeneous spaces Ann. Math. 76 (1962), 446–463. Google Scholar

[23] 23.Pierce, R. S., Associative algebras, Graduate Texts in Mathematics, vol. 88. Studies in the History of Modern Science, 9. (Springer-Verlag, New York-Berlin, 1982). Google Scholar

[24] 24.Ronan, M. A., Triangle geometries, J. Comb. Theory Ser. A 37 (1984), 294–319. Google Scholar | DOI

[25] 25.Ronan, M. A., Lectures on buildings, Perspectives in Mathematics, vol. 7. (Academic Press, Inc., Boston, MA, 1989). Google Scholar

[26] 26.Raghunathan, M. C., Discrete subgroups of algebraic groups over local fields of positive characteristics, Proceedings of the Indian Academy of Science (math. sci.) 99 (2) (1989), 127–146. Google Scholar

[27] 27.Salehi Golsefidy, A., Lattices of minimum covolume in Chevalley groups over local fields of positive characteristics, Duke Math. J. 146 (2) (2009), 227–251. Google Scholar

[28] 28.Tits, J., Buildings of spherical type and finite BN–pairs, Lecture Notes in Mathematics, vol. 386, (Springer-Verlag, New York, 1974). Google Scholar

[29] 29.Tits, J., Endliche Spielungsgruppen, die als Weylgruppen auftreten, Invent. Math. (1977), 283–295. Google Scholar

[30] 30.Tits, J., Immeubles de type affine, Buildings and the geometry of diagrams (Rosati, L. A., Editor) Lecture Notes in Mathematics, vol. 1181 (Springer-Verlag, Berlin, 1986) 159–190. Google Scholar | DOI