Introduction to Λ-Trees

Chiswell, I. M.

doi:10.1007/978-94-011-0149-3_9

Cited by 22 publications

(48 citation statements)

References 28 publications

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“…If g is a hyperbolic isometry, lðgÞ is defined to be the amplitude of the translation on its axis, and if g is elliptic or an inversion, lðgÞ is defined to be 0. Thus lðgÞ A L, and is called the hyperbolic length of g. (See [3,Chapter 3]. )…”

Section: The Case Of a Tree Of Groupsmentioning

confidence: 99%

“…Since a group is right orderable if and only if every finitely generated subgroup is right orderable, it su‰ces to show this under the extra hypothesis that G is finitely generated, and this will be done by induction on n. For n ¼ 1, all groups acting freely on R-trees are (two-sided) orderable. See [3, Proposition 5.13, Chapter 5] and the remarks following it in [3]. Now assume that it is true for n À 1, and let G be a finitely generated group acting freely on an R n -tree.…”

Section: The Case Of a Tree Of Groupsmentioning

confidence: 99%

“…The corollary applies when L is a finitely generated ordered abelian group (see the remarks in [3] preceding [3, Lemma 1.6, Chapter 1]). A special case is when L ¼ Z n with the lexicographic order.…”

Section: The Case Of a Tree Of Groupsmentioning

confidence: 99%

“…At the end of Chapter 5 in [3] it is asked whether or not all tree-free groups are orderable, or at least right orderable. The context is that groups in the class of locally fully residually free groups, a large class of tree-free groups, are (two-sided) orderable.…”

Section: Introductionmentioning

confidence: 99%

“…The subtree Y spanned by the G-orbit of x 0 is 6 g A G ½x 0 ; gx 0 (see the definition in [3] preceding [3, Lemma 1.8, Chapter 2]), and is G-invariant. If y; z A Y then dðy; zÞ c dðx 0 ; yÞ þ dðx 0 ; zÞ, hence dðy; zÞ A R n .…”

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Right orderability and graphs of groups

Chiswell

2011

Journal of Group Theory

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Section: The Case Of a Tree Of Groupsmentioning

confidence: 99%

Section: The Case Of a Tree Of Groupsmentioning

confidence: 99%

Section: The Case Of a Tree Of Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Right orderability and graphs of groups

Chiswell

2011

Journal of Group Theory

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Limit groups as limits of free groups

2005

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We give a topological framework for the study of Sela's limit groups: limit groups are limits of free groups in a compact space of marked groups. Many results get a natural interpretation in this setting. The class of limit groups is known to coincide with the class of finitely generated fully residually free groups. The topological approach gives some new insight on the relation between fully residually free groups, the universal theory of free groups, ultraproducts and non-standard free groups.

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Compactness and local compactness for $${\mathbb{R}}$$ -trees

2008

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We establish (geometric) criteria for an R-tree to be compact and to be locally compact. It follows that locally compact R-trees are separable. Mathematics Subject Classification (2000). Primary 20E08; Secondary 54E45, 54E50.The purpose of this note is to provide criteria for an R-tree to be compact and to be locally compact. These are expressed entirely in terms of the geometry of the tree. Compact and locally compact R-trees have been used in probability theory; see, for example, [2] and the references cited there. A general reference for the theory of R-trees is [1]. As usual, [x, y] denotes the segment joining points x, y in an R-tree. The ideas of linear subtree, ray and end are discussed in [1, Chapter 2, §4], and for the notions of direction, degree of a point and endpoint see [1] after Lemma 1.7 in Chapter 2.To state the compactness criterion, the following two definitions are needed. Let (X, d) be an R-tree, and let r > 0 be a real number.The second definition uses the idea of projection of a point onto a segment. For a discussion of projection, see [1, Chapter 2, Lemma 1.9] and the remarks following it. The projection of a point z onto a segment [x, y] is simply the point Y(x, y, z) defined after Chapter 2, Lemma 1.2 in [1].

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Introduction to Λ-Trees

Cited by 22 publications

References 28 publications

Right orderability and graphs of groups

Right orderability and graphs of groups

Limit groups as limits of free groups

Compactness and local compactness for $${\mathbb{R}}$$ -trees

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