Introduction to Λ-Trees

Chiswell, I. M.

doi:10.1142/9789812810533

Cited by 45 publications

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“…1 We use induction on t := r + s ≥ 2. If t = 2, then r = s = 1, so there is nothing to prove in this case.…”

Section: Proofmentioning

confidence: 99%

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A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

Müller

2010

Abh. Math. Semin. Univ. Hambg.

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This paper continues the investigation of the groups RF (G) first introduced in the forthcoming book of Chiswell and Müller "A Class of Groups Universal for Free R-Tree Actions" and in the article by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193-227, 2009). We establish a criterion for a family {H σ } of hyperbolic subgroups H σ ≤ RF (G) to generate a hyperbolic subgroup isomorphic to the free product of the H σ (Theorem 1.2), as well as a local-global principle for local incompatibility (Theorem 4.1). In conjunction with the theory of test functions as developed by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193-227, 2009), these results allow us to obtain a necessary and sufficient condition for a free product of real groups to embed as a hyperbolic subgroup in RF (G) for a given group G (Corollary 5.4). As a further application, we show that the centralizers associated with a family of pairwise locally incompatible cyclically reduced functions in RF (G) generate a hyperbolic subgroup isomorphic to the free product of these centralizers (Corollary 5.2).

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“…1 We use induction on t := r + s ≥ 2. If t = 2, then r = s = 1, so there is nothing to prove in this case.…”

Section: Proofmentioning

confidence: 99%

“…The family of generating sets {E σ } σ ∈S clearly satisfies (1), since the f σ are assumed to be pairwise locally incompatible; hence, by Theorem 5.1, the subgroup generated by the centralizers C RF (G) (f σ ) is hyperbolic, and is isomorphic to the free product of these centralizers. Summarizing, we have obtained the following.…”

Section: Some Applicationsmentioning

confidence: 99%

A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

Müller

2010

Abh. Math. Semin. Univ. Hambg.

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“…Easy to prove in the tree case, see for example [6], the generalization to -affine buildings turns out to be much harder.…”

Section: Introductionmentioning

confidence: 99%

Λ-buildings and base change functors

Schwer¹,

Struyve

2011

Geom Dedicata

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We prove an analog of the base change functor of Lambda-trees in the setting of generalized affine buildings. The proof is mainly based on local and global combinatorics of the associated spherical buildings. As an application we obtain that the class of generalized affine buildings is closed under taking ultracones and asymptotic cones. Other applications involve a complex of groups decompositions and fixed point theorems for certain classes of generalized affine buildings

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“…The study of isometric actions on real trees has had some success, notably Rips' Theorem which characterizes finitely generated groups that admit a free isometric action on a real tree ( [7], [8], [4]), and it has become an established technique in geometric group theory.…”

Section: Introductionmentioning

confidence: 99%

“…While several authors have extended many of the results from the theory of real trees to that of L-trees ( [4] provides a comprehensive exposition) this has not been as widely adopted as a technique in the study of discrete groups as has the theory of real trees. This is due in part to an incomplete understanding of free actions on L-trees, and in part to a perceived lack of motivation in passing from R to a more general L.…”

Section: Introductionmentioning

confidence: 99%

Some free actions on non-archimedean trees

Martino¹,

Rourke²

2004

Journal of Group Theory

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In this paper we show that various groups are Z n -free. In particular we show that almost every surface group is ðZ Â ZÞ-free as are the groups of Liousse [11]. We also demonstrate that the class of Z n -free groups is closed under taking amalgamated free products over an infinite cyclic group as long as it is maximal abelian in each vertex group. It follows that a large class of hyperbolic groups are Z n -free.

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Introduction to Λ-Trees

Cited by 45 publications

References 0 publications

A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

Λ-buildings and base change functors

Some free actions on non-archimedean trees

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