COCOMPACT LATTICES ON Ãn BUILDINGS

Inna Capdeboscq; Dmitriy Rumynin; Anne Thomas

doi:10.1017/S0017089514000287

COCOMPACT LATTICES ON Ã_n BUILDINGS

Inna Capdeboscq
, Dmitriy Rumynin
, Anne Thomas

University of Warwick
University of Glasgow

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We construct cocompact lattices Γ′₀ < Γ₀ in the group G = PGL_d (double-struck 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 PGL_d (q). We show that the intersections of Γ′₀ and Γ₀ with PSL_d (double-struck F_q((t))) are lattices in PSL_d(double-struck F_q((t))), and identify the pairs (d, q) such that the entire lattice Γ′₀ or Γ₀ is contained in PSL_d(double-struck F_q((t))). Finally we discuss minimality of covolumes of cocompact lattices in SL₃(double-struck 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.

Original language	English
Pages (from-to)	241-262
Number of pages	22
Journal	Glasgow Mathematical Journal
Volume	57
Issue number	2
DOIs	https://doi.org/10.1017/S0017089514000287
Publication status	Published - 20 May 2015
Externally published	Yes

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title = "COCOMPACT LATTICES ON {\~A}n BUILDINGS",

abstract = "We construct cocompact lattices Γ′0 < Γ0 in the group G = PGLd (double-struck Fq((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 (double-struck Fq((t))) are lattices in PSLd(double-struck Fq((t))), and identify the pairs (d, q) such that the entire lattice Γ′0 or Γ0 is contained in PSLd(double-struck Fq((t))). Finally we discuss minimality of covolumes of cocompact lattices in SL3(double-struck Fq((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.",

author = "Inna Capdeboscq and Dmitriy Rumynin and Anne Thomas",

note = "Publisher Copyright: Copyright {\textcopyright} Glasgow Mathematical Journal Trust 2014.",

year = "2015",

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doi = "10.1017/S0017089514000287",

language = "English",

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AU - Capdeboscq, Inna

AU - Rumynin, Dmitriy

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N2 - We construct cocompact lattices Γ′0 < Γ0 in the group G = PGLd (double-struck Fq((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 (double-struck Fq((t))) are lattices in PSLd(double-struck Fq((t))), and identify the pairs (d, q) such that the entire lattice Γ′0 or Γ0 is contained in PSLd(double-struck Fq((t))). Finally we discuss minimality of covolumes of cocompact lattices in SL3(double-struck Fq((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.

AB - We construct cocompact lattices Γ′0 < Γ0 in the group G = PGLd (double-struck Fq((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 (double-struck Fq((t))) are lattices in PSLd(double-struck Fq((t))), and identify the pairs (d, q) such that the entire lattice Γ′0 or Γ0 is contained in PSLd(double-struck Fq((t))). Finally we discuss minimality of covolumes of cocompact lattices in SL3(double-struck Fq((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.

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DO - 10.1017/S0017089514000287

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VL - 57

SP - 241

EP - 262

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

IS - 2

ER -

COCOMPACT LATTICES ON Ã_n BUILDINGS

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