Cocompact lattices on A~_n buildings

Capdeboscq, I., Rumynin, D. and Thomas, A. (2015) Cocompact lattices on A~_n buildings. Glasgow Mathematical Journal, 57(2), pp. 241-262. (doi: 10.1017/S0017089514000287)

Abstract

We construct cocompact lattices Γ’₀< Γ₀ in the group G = PGL_d(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 [CS]. 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 PGL_d(q). We show that the intersections of Γ’₀ and Γ₀ with PSL_d(F_q((t))) are lattices in PSL_d(F_q((t))), and identify the pairs (d; q) such that the entire lattice Γ’₀ or Γ₀ is contained in PSL_d(F_q((t))). Finally we discuss minimality of covolumes of cocompact lattices in SL₃(F_q((t))). Our proofs combine the construction of Cartwright{Steger [CS] with results about Singer cycles and their normalisers, and geometric arguments.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Thomas, Dr Anne and Rumynin, Dr Dmitrity
• Authors: Capdeboscq, I., Rumynin, D., and Thomas, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics
• Journal Name: Glasgow Mathematical Journal
• Publisher: Cambridge University Press
• ISSN: 0017-0895
• ISSN (Online): 1469-509X
• Copyright Holders: Copyright © Cambridge University Press
• First Published: First published in Glasgow Mathematical Journal 57(2):241-262
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher.