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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.

MSC:

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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References:

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