Lattices in product of trees. (English) Zbl 1007.22013

The authors study the structure of lattices in products \(\text{Aut }T_1\times \text{Aut }T_2\) of automorphism groups of regular trees. These lattices have a rich structure theory parallel to the theory of lattices in semisimple Lie groups and exhibiting some new phenomena. In Chapter 1 the authors show that the object corresponding to torsion-free, discrete subgroups of \(\text{Aut }T_1\times \text{Aut }T_2\) is square complex, with additional structure. In Chapter 2 they consider irreducible cocompact lattices with locally quasi-primitive projections. In Chapter 3 they obtain certain cohomological vanishing results for irreducible lattices with locally quasi-primitive projections. In Chapter 4 they prove that every nontrivial normal subgroup of a lattice whose projections satisfy stronger transitivity conditions is of finite index. In Chapter 5 they produce effective sufficient conditions on a finite square complex ensuring that its fundamental group is of a certain natural type. In Chapter 6 they construct, for every \(n\geq 15\), \(m\geq 19\), a square complex \(X_{n,m}\) on one vertex whose fundamental group has dense projections.


22D12 Other representations of locally compact groups
22E40 Discrete subgroups of Lie groups
20G35 Linear algebraic groups over adèles and other rings and schemes
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