Let T be a locally finite tree all of whose vertices have valency at least 6. We classify, up to isomorphism, the closed subgroups of Aut(T ) acting 2-transitively on the set of ends of T and whose local action at each vertex contains the alternating group. The outcome of the classification for a fixed tree T is a countable family of groups, all containing two remarkable subgroups: a simple subgroup of index ≤ 8 and (the semiregular analog of) the universal locally alternating group of Burger-Mozes (with possibly infinite index). We also provide an explicit example showing that the statement of this classification fails for trees of smaller degree.
Let T be a locally finite tree without vertices of degree 1. We show that among the closed subgroups of Aut(T ) acting with a bounded number of orbits, the Chabautyclosure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of T have degree ≥ 3, then the set of isomorphism classes of topologically simple closed subgroups of Aut(T ) acting doubly transitively on ∂T carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.
Let Γ ≤ Aut(T d 1 ) × Aut(T d 2 ) be a group acting freely and transitively on the product of two regular trees of degree d 1 and d 2 . We develop an algorithm which computes the closure of the projection of Γ on Aut(T dt ) under the hypothesis that d t ≥ 6 is even and that the local action of Γ on T dt contains Alt(d t ). We show that if Γ is torsion-free and d 1 = d 2 = 6, exactly seven closed subgroups of Aut(T 6 ) arise in this way. We also construct two new infinite families of virtually simple lattices in Aut(T 6 ) × Aut(T 4n ) and in Aut(T 2n ) × Aut(T 2n+1 ) respectively, for all n ≥ 2. In particular we provide an explicit presentation of a torsion-free infinite simple group on 5 generators and 10 relations, that splits as an amalgamated free product of two copies of F 3 over F 11 . We include information arising from computerassisted exhaustive searches of lattices in products of trees of small degrees. In an appendix by Pierre-Emmanuel Caprace, some of our results are used to show that abstract and relative commensurator groups of free groups are almost simple, providing partial answers to questions of Lubotzky and Lubotzky-Mozes-Zimmer.
We build a building of typeà 2 and a discrete group of automorphisms acting simply transitively on its set of vertices. The characteristic feature of this building is that its rank 2 residues are isomorphic to the Hughes projective plane of order 9, which is non-Desarguesian. This solves a problem asked by W. Kantor in 1986, as well as a question asked by J. Howie in 1989.
We prove a purely topological characterization of the Moufang property for disconnected compact polygons in terms of convergence groups. As a consequence, we recover the fact that a locally finite thick affine building of rank 3 is a Bruhat-Tits building if and only if its automorphism group is strongly transitive. We also study automorphism groups of general compact polygons without any homogeneity assumption. A compactness criterion for sets of automorphisms is established, generalizing the theorem by Burns and Spatzier that the full automorphism group, endowed with the compact-open topology, is a locally compact group.
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