Introduction to Λ-Trees

Chiswell, I. M.

doi:10.1142/4495

Cited by 76 publications

(12 citation statements)

References 31 publications

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“…and by Lemma 2.1.4 in [8] we have that That is, [a, e a , e b , b] is a piecewise segment Proof. Follows from Lemma 2.4 and Lemma 2.5.…”

Section: 3mentioning

confidence: 91%

“…In an R-tree, [a, b] denotes the unique geodesic segment from a to b. Since metric structures are required to be based on complete metric spaces, it is a helpful fact that the completion of an R-tree is an R-tree (see [8,Lemma 2.4.14]).…”

Section: 3mentioning

confidence: 99%

“…The following lemmas and definitions collect some straightforward facts about R-trees used in this paper. For helpful pictures and more facts about R-trees see [8]. Moreover, for all b ∈ E 1 and c ∈ E 2 , the geodesic segment from b to c must contain [u, v].…”

Section: 3mentioning

confidence: 99%

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Model Theory of R-trees

Carlisle,

Henson

2018

Preprint

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We show the theory of pointed R-trees with radius at most r is axiomatizable in a suitable continuous signature. We identify the model companion rbRTr of this theory and study its properties. In particular, the model companion is complete and has quantifier elimination; it is stable but not superstable. We identify its independence relation and find built-in canonical bases for non-algebraic types. Among the models of rbRTr are R-trees that arise naturally in geometric group theory. In every infinite cardinal, we construct the maximum possible number of pairwise non-isomorphic models of rbRTr; indeed, the models we construct are pairwise non-homeomorphic. We give detailed information about the type spaces of rbRTr. Among other things, we show that the space of 2-types over the empty set is nonseparable. Also, we characterize the principal types of finite tuples (over the empty set) and use this information to conclude that rbRTr has no atomic model.

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“…and by Lemma 2.1.4 in [8] we have that That is, [a, e a , e b , b] is a piecewise segment Proof. Follows from Lemma 2.4 and Lemma 2.5.…”

Section: 3mentioning

confidence: 91%

Section: 3mentioning

confidence: 99%

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Model Theory of R-trees

Carlisle,

Henson

2018

Preprint

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“…Then it is easy to see that we may take w = ghz. It is impossible that both rules are of type (2), so assume one is of type (1) and the other is of type (2). There are two possibilities; the first is that the two rules have the form…”

Section: The Rewriting System Rmentioning

confidence: 99%

“…By a tree-free group we mean a group having an action on a -tree, for some (totally) ordered abelian group , which is free and without inversions. This means that every nonidentity element acts as a hyperbolic isometry (see [2,Chapter 3]). This paper arose from our efforts to verify some of the results in the paper of Alperin and Moss [1], one of the seminal papers on R-trees.…”

Section: Introductionmentioning

confidence: 99%

Embedding theorems for tree-free groups

Chiswell

Müller

2010

Math. Proc. Camb. Phil. Soc.

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We establish two embedding theorems for tree-free groups. The first result embeds a group G acting freely and without inversions on a Λ-tree X into a group acting freely, without inversions, and transitively on a Λ-tree in such a way that X embeds into by means of a G-equivariant isometry. The second result embeds a group G acting freely and transitively on an ℝ-tree X into (H) for some suitable group H, again in such a way that X embeds G-equivariantly into the ℝ-tree XH associated with (H). The group (H) referred to here belongs to a class of groups introduced and studied by the present authors in [3]. As a consequence of these two theorems, we find that -groups and their associated ℝ-trees are in fact universal for free ℝ-tree actions. Moreover, our first embedding theorem throws light on the question, arising from the results of [7], whether a group endowed with a Lyndon length function L can always be embedded in a length-preserving way into a group with a regular Lyndon length function; modulo an obvious necessary restriction we show that this is indeed the case if L is free.

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Abelianization and fixed point properties of units in integral group rings

Bächle

Janssens

Jespers

et al. 2022

Mathematische Nachrichten

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Let 𝐺 be a finite group and  (ℤ𝐺) the unit group of the integral group ring ℤ𝐺. We prove a unit theorem, namely, a characterization of when  (ℤ𝐺) satisfies Kazhdan's property (T), both in terms of the finite group 𝐺 and in terms of the simple components of the semisimple algebra ℚ𝐺. Furthermore, it is shown that for  (ℤ𝐺), this property is equivalent to the weaker property FAb (i.e., every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property FA, denoted HFA. More precisely, it is described when all subgroups of finite index in  (ℤ𝐺) have both finite abelianization and are not a nontrivial amalgamated product. A crucial step for this is a reduction to arithmetic groups SL 𝑛 (), where  is an order in a finite-dimensional semisimple ℚ-algebra 𝐷, and finite groups 𝐺, which have the so-called cut property. For such groups 𝐺, we describe the simple epimorphic images of ℚ𝐺. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups E 𝑛 (𝐷) of SL 𝑛 (𝐷). These groups are well understood except in the degenerate case of lower rank, that is, for SL 2 () with  an order in a division algebra 𝐷 with a finite number of units. In this setting, we determine Serre's property FA for E 2 () and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its ℤ-rank.

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Introduction to Λ-Trees

Cited by 76 publications

References 31 publications

Model Theory of R-trees

Model Theory of R-trees

Embedding theorems for tree-free groups

Abelianization and fixed point properties of units in integral group rings

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