Cocompact lattices on A_n buildings

2012/06/23 by Inna Capdeboscq, Capdeboscq, Inna, Dmitriy Rumynin +3
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1206.5356

openalex publication_date 2012/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be the field of formal Laurent series over the finite field of order q. We construct cocompact lattices Γ'₀ < Γ₀ in the group G = PGL_d(K) which are type-preserving and act transitively on the set of vertices of each type in the building associated to G. 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 then show that the intersections of Γ'₀ and Γ₀ with PSL_d(K) are lattices in PSL_d(K), and identify the pairs (d,q) such that the entire lattice Γ'₀ or Γ₀ is contained in PSL_d(K). Finally we discuss minimality of covolumes of cocompact lattices in SL₃(K). Our proofs combine a construction of Cartwright and Steger with results about Singer cycles and their normalisers, and geometric arguments.

Citations

• Arithmetic Subgroups of Algebraic Groups

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