Burger, Marc; Mozes, Shahar Lattices in product of trees. (English) Zbl 1007.22013 Publ. Math., Inst. Hautes Étud. Sci. 92, 151-194 (2000). 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. Reviewer: B.M.Schein (Fayetteville) Cited in 3 ReviewsCited in 104 Documents 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 Keywords:lattices; automorphism groups of regular trees; irreducible cocompact lattices × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] A. E. Brouwer, A. M. Cohen, A. Neumaier,Distance Regular Graphs, Ergebnisse, 3. Folge, Band 18, Springer 1989. · Zbl 0747.05073 [2] M. Burger, S. Mozes, CAT(-1)-spaces, divergence groups and their commensurators,J. Amer. Math. 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Kirby; http://math.berkeley.edu/\(\sim\)kirby/. · Zbl 0892.57013 [16] Ch. E. Praeger, Finite quasiprimitive graphs inSurveys in Combinatorics, 1997, Proc. of the 16th British combinatorial conference, London UK, July 1997. London: Cambridge University Press,Lond. Math. Soc. L.N.S. 241 (1997), 65–85. [17] J.-P. Serre,Trees, Springer 1980. [18] J. G. Thompson, Bounds for orders of maximal subgroups,J. Algebra 14 (1970), 135–138. · Zbl 0221.20031 · doi:10.1016/0021-8693(70)90117-1 [19] D. Wise, A non-positively curved squared complex with no finite covers (1995), Preprint. [20] H. Wielandt,Finite Permutation Groups, Academic Press (1964). · Zbl 0138.02501 [21] H. Wielandt, Subnormal subgroups and permutation groups,Ohio State University Lecture Notes, Columbus (1971). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. 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