Quasi-automorphisms of the infinite rooted 2-edge-coloured binary tree

Nucinkis, Brita E. A.; John-Green, Simon St

doi:10.4171/ggd/448

Cited by 6 publications

(10 citation statements)

References 17 publications

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“…By definition piecewise planar-tree isometric vertex permutations respect not only the pairs of end points of almost all edges of T but also the cyclic ordering of the stars at almost all vertices of T . Hence g can also be referred to as a quasi-(planar-tree) automorphism of the spine T -see ( [Leh08], [LS07], [BMN13], and [NSt15]).…”

Section: 2mentioning

confidence: 99%

“…Thus, progress in this direction seems accessible -whether pei(S)/sym(S) is better behaved than pei(S) itself remains to be seen. For the Houghton groups this is trivially true, and the subtle difference between QV and QV (see [NSt15]) might indicate that this is indeed the case. iii) Defining and studying a group pal(Z n ) of piecewise afine-linear permutations on Z n , and find the footprints of the structure of SL n (Z n ) in its structure.…”

Section: Introductionmentioning

confidence: 95%

“…This exhibits G Γ * (Ω) as a permutation group on the vertices of the tree T . We discuss whether this action respects almost all edges and cyclic star-orderings of T (following the terminology of [Leh08], [LS07], [BMN13], and [NSt15], this would be an action by quasi planar-tree automorphisms).…”

Section: Introductionmentioning

confidence: 99%

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Groups of piecewise isometric permutations of lattice points or finitary rearrangements of tessellations

Bieri¹,

Sach²

2021

Preprint

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Through the glasses of didactic reduction: We consider a (periodic) tessellation ∆ of either Euclidean or hyperbolic n-space M . By a piecewise isometric rearrangement of ∆ we mean the process of cutting M along corank-1 tile-faces into finitely many convex polyhedral pieces, and rearranging the pieces to a new tight covering of the tessellation ∆. Such a rearrangement defines a permutation of the (centers of the) tiles of ∆, and we are interested in the group P I(∆) of all piecewise isometric rearrangements of ∆.In this paper we offer: a) An illustration of piecewise isometric rearrangements in the visually attractive hyperbolic plane, b) an explanation how this is related to Richard Thompson's groups, c) a chapter on the structure of the group pei(Z n ) of all piecewise Euclidean rearrangements of the standard tessellation of R n by unitcubes, and d) results on the finiteness properties of some subgroups of pei(Z n ). ContentsChapter 1. Introduction 3 1. Generalities and main result 3 Chapter 2. On the hyperbolic case 11 2. Planar hyperbolic examples 11 Chapter 3. The Euclidean case I: The structure of pei(S) 20 3. Orthohedral sets 20 4. Permutation groups supported on orthohedral sets 24

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Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

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Groups of piecewise isometric permutations of lattice points or finitary rearrangements of tessellations

Bieri¹,

Sach²

2021

Preprint

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“…We denote by QV n,r,p the quasi-automorphism group of the union T = T n,r,p of r infinite rooted n-regular trees with p isolated vertices (that is, p rooted trees without children), ie., the group of bijections T (0) → T (0) between the vertices of T which preserve adjacency and the left-to-right ordering of the children of a given vertex for all but finitely many vertices. In particular, QV 2,0,0 and QV 2,0,1 are respectively the groups QV and QV studied in [NSJG18]…”

Section: Examples Of Braided Diagram Groupsmentioning

confidence: 99%

Embeddings into Thompson's groups from quasi-median geometry

Genevois

2019

Groups Geom. Dyn.

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The main result of this article is that any braided (resp. annular, planar) diagram group D splits as a short exact sequence 1 → R → D → S → 1 where R is a subgroup of some right-angled Artin group and S a subgroup of Thompson's group V (resp. T , F ). As an application, we show that several braided diagram groups embeds into Thompson's group V , including Higman's groups Vn,r, groups of quasi-automorphisms QVn,r,p, and generalised Houghton's groups Hn,p. arXiv:1709.03888v2 [math.GR]

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“…It follows that every piecewise Γ-isometric permutation of Ω = ver(T ) can be interpreted as a piecewise planar tree isometric (ppti) permutation of the vertices of the tree T -i.e, a permutation of ver(T ) that respect all but finitely many edges and the link-orientations at all but finitely many vertices of T . Other authors use the term quasi-autmorphisms ([Leh08], [LS07], [BMN13], and [NSt15]). Conversely, the convex closure of T 1 (p) is a convex Γ-polyhedral subset P with ver(T 1 (p)) = Ω ∩ P , and the tree-isometric embedding T 1 (p) → T extends to an isometric embedding of the convex closure of T 1 (p) into H 2 .…”

Section: A Hyperbolic Examplementioning

confidence: 99%

Groups of piecewise isometric permutations of lattice points

Bieri,

Sach

2016

Preprint

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Let M denote either Euclidean or hyperbolic N -space, and let Γ be a discrete group of isometries of M , with the property that Γ respects and acts tile-transitively on a convex polyhedral tesselation of M . Given an arbitrary base point p ∈ M , we consider the orbit Ω := Γp ⊆ M and define a notion of "Γ-polyhedral pieces" S ⊆ Ω. The objects of our interest are the groups G Γ (S) of all piecewise Γ-isometric permutations on S. In this paper we merely present the two most basic examples, and these play rather different roles: The case when Γ = PSL 2 (Z) acting on the hyperbolic plane reveals that the groups G Γ (Ω) here have prominent relatives: they are closely related to Richard Thompson's group V . And in the case Γ = Isom(Z N ) we find that the groups G Γ (S) have diverse but to some extent computable finiteness properties. The conjunction of the two examples suggests that to investigate the piecewise Γ-isometric permutation groups more systematically might be a worthwhile project and might yield interesting new groups with an accessible finiteness pattern.

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Quasi-automorphisms of the infinite rooted 2-edge-coloured binary tree

Cited by 6 publications

References 17 publications

Groups of piecewise isometric permutations of lattice points or finitary rearrangements of tessellations

Groups of piecewise isometric permutations of lattice points or finitary rearrangements of tessellations

Embeddings into Thompson's groups from quasi-median geometry

Groups of piecewise isometric permutations of lattice points

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