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COCOMPACT LATTICES ON (A)over-tilde(n) BUILDINGS

Author:
Capdeboscq, Inna  Rumynin, Dmitriy  Thomas, Anne  


Journal:
GLASGOW MATHEMATICAL JOURNAL


Issue Date:
2015


Abstract(summary):

We construct cocompact lattices Gamma(')(0) < Gamma(0) in the group G =3D PGL(d)(F-q((t))) which are type-preserving and act transitively on the set of vertices of each type in the building Delta 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 Gamma(')(0) is a Singer cycle and the stabiliser of each vertex in Gamma(0) is isomorphic to the normaliser of a Singer cycle in PGL(d)(q). We show that the intersections of Gamma(')(0) and Gamma(0) with PSLd(Fq((t))) are lattices in PSLd (F-q((t))), and identify the pairs (d, q) such that the entire lattice Gamma(')(0) or Gamma(0) is contained in PSLd(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 Isr. J. Math. 103 (1998), 125-140 with results about Singer cycles and their normalisers, and geometric arguments.


Page:
241---262


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