Geodesic
COCOMPACT LATTICES ON Ãn BUILDINGS
Glasgow mathematical journal, Tome 57 (2015) no. 2, pp. 241-262

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DOI

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. 
CAPDEBOSCQ, INNA; RUMYNIN, DMITRIY; THOMAS, ANNE. COCOMPACT LATTICES ON Ãn BUILDINGS. Glasgow mathematical journal, Tome 57 (2015) no. 2, pp. 241-262. doi: 10.1017/S0017089514000287
@article{10_1017_S0017089514000287,
     author = {CAPDEBOSCQ, INNA and RUMYNIN, DMITRIY and THOMAS, ANNE},
     title = {COCOMPACT {LATTICES} {ON} {\~An} {BUILDINGS}},
     journal = {Glasgow mathematical journal},
     pages = {241--262},
     year = {2015},
     volume = {57},
     number = {2},
     doi = {10.1017/S0017089514000287},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000287/}
}
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