Groups acting on trees: From local to global structure

Burger, Marc; Mozes, Shahar

doi:10.1007/bf02698915

Cited by 156 publications

(260 citation statements)

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“…They arise as lattices in products of automorphism groups of regular trees, which are new classes of locally compact groups, for example, are not Lie groups. Briefly, they are given by certain lattices of Aut T 1 × Aut T 2 whose projections in each factor satisfy various transitivity conditions, in particular, the so-called locally quasiprimitive condition [BuM2]. Another new feature of such lattices in comparison with lattices in semisimple Lie groups is that that cocompact lattices in Aut T 1 × Aut T 2 never have dense projections in the two factors.…”

Section: "One Of Main Applications Of the Normal Subgroup Theorem Is mentioning

confidence: 99%

A Summary of the Work of Gregory Margulis

Ji¹

2008

Pure and Applied Mathematics Quarterly

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Section: "One Of Main Applications Of the Normal Subgroup Theorem Is mentioning

confidence: 99%

A Summary of the Work of Gregory Margulis

Ji¹

2008

Pure and Applied Mathematics Quarterly

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“…Since non-trivial elements of FSym(V) cannot be in U, the subgroup FSym(V) is closed in this topology and is easily seen to be normal as well. (It may be seen that FSym(V) is the quasi-centre of the group of almost automorphisms, see [32] for the definition.) The quotient of the group of almost automorphisms by FSym(V) is therefore a locally compact group which will be denoted by AAut(T ).…”

Section: Glöckner Bases This Decomposition On a Variation On ([18] Lementioning

confidence: 99%

Computing the Scale of an Endomorphism of a totally Disconnected Locally Compact Group

Willis

2017

Axioms

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Abstract:The scale of an endomorphism of a totally disconnected, locally compact group G is defined and an example is presented which shows that the scale function is not always continuous with respect to the Braconnier topology on the automorphism group of G. Methods for computing the scale, which is a positive integer, are surveyed and illustrated by applying them in diverse cases, including when G is compact; an automorphism group of a tree; Neretin's group of almost automorphisms of a tree; and a p-adic Lie group. The information required to compute the scale is reviewed from the perspective of the, as yet incomplete, general theory of totally disconnected, locally compact groups.

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“…It is wellknown that simplicial trees are fundamental in combinatorial group theory (see [Se2]). For a survey of more recent developments and generalizations, see the book [BL] and [BuM1] [BuM2]. Many natural discrete subgroups in the automorphism group of trees have been constructed from Kac-Moody groups and subgroups.…”

Section: Finite Groupsmentioning

confidence: 99%

Buildings and their Applications in Geometry and Topology

Ji¹

2006

Asian Journal of Mathematics

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Abstract. Buildings were first introduced by J. Tits in 1950s to give systematic geometric interpretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (Bruhat-Tits buildings), topological buildings, R-buildings, in particular R-trees. They are useful for many different applications in various subjects: algebraic groups, finite groups, finite geometry, representation theory over local fields, algebraic geometry, Arakelov intersection for arithmetic varieties, algebraic K-theories, combinatorial group theory, global geometry and algebraic topology, in particular cohomology groups, of arithmetic groups and S-arithmetic groups, rigidity of cofinite subgroups of semisimple Lie groups and nonpositively curved manifolds, classification of isoparametric submanifolds in R n of high codimension, existence of hyperbolic structures on three dimensional manifolds in Thurston's geometrization program. In this paper, we survey several applications of buildings in differential geometry and geometric topology. There are four underlying themes in these applications:1. Buildings often describe the geometry at infinity of symmetric spaces and locally symmetric spaces and also appear as limiting objects under degeneration or scaling of metrics. 2. Euclidean buildings are analogues of symmetric spaces for semisimple groups defined over local fields and their discrete subgroups. 3. Buildings of higher rank are rigid and hence objects which contain or induce higher rank buildings tend to be rigid. 4. Additional structures on buildings, for example, topological buildings, are important in applications for infinite groups.

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Groups acting on trees: From local to global structure

Cited by 156 publications

References 20 publications

A Summary of the Work of Gregory Margulis

A Summary of the Work of Gregory Margulis

Computing the Scale of an Endomorphism of a totally Disconnected Locally Compact Group

Buildings and their Applications in Geometry and Topology

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