Higher Dimensional Thompson Groups

Abstract

We construct a "higher dimensional" version 2V of Thompson's group V. Like V it is an infinite, finitely presented, simple subgroup of the homeomorphism group of the Cantor set, but we show that it is not isomorphic to V by showing that the actions on the Cantor set are not topologically conjugate: 2V has an element with "chaotic" action, while V cannot have such an element. A theorem of Rubin is then applied which shows that for these two groups, isomorphism would imply topological conjugacy.

27 pages To appear in Geometriae Dedicata

Cited by in corpus (44)

• The Algebra of Strand Splitting. I. A Braided Version of Thompson's Group V
• On the Baker's map and the Simplicity of the Higher Dimensional Thompson Groups nV
• Presentations of higher dimensional Thompson groups
• The homology of the Higman-Thompson groups
• Embeddings into Thompson's group $V$ and $co\mathcal{C}\mathcal{F}$ groups
• Topological Full Groups of Ample Groupoids with Applications to Graph Algebras
• The Brin-Thompson groups sV are of type F_\infty
• A Thompson group for the basilica
• Asymptotically rigid mapping class groups I: Finiteness properties of braided Thompson's and Houghton's groups
• Twisted Brin-Thompson groups
• Isomorphisms of Brin-Higman-Thompson groups
• On the distortion of twin building lattices
• Classifying spaces from Ore categories with Garside families
• Rigidity properties of full groups of pseudogroups over the Cantor set
• $L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ does not embed in $L_{2,\mathbb{Z}}$
• Diffeomorphisms groups of tame Cantor sets and Thompson-like groups
• The Neretin groups
• Cohomological finiteness conditions and centralisers in generalisations of Thompson's group V
• On groups whose actions on finite-dimensional CAT(0) spaces have global fixed points
• Infinite $\frac{3}{2}$-generated groups
• A finitely presented infinite simple group of homeomorphisms of the circle
• A short proof of Rubin's theorem
• Presentations of generalisations of Thompson's group $V$
• Etale groupoids arising from products of shifts of finite type
• Diffeomorphism groups of critical regularity
• Braiding groups of automorphisms and almost-automorphisms of trees
• Braided Brin-Thompson Groups
• A Class of Rearrangement Groups that are not Invariably Generated
• Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces
• Higher dimensional Thompson groups have Serre's property FA
• Generalizations of Loday's assembly maps for Lawvere's algebraic theories
• Embeddings of right-angled Artin groups into higher dimensional Thompson groups
• Hyperbolic and cubical rigidities of Thompson's group V
• L2-invisibility of symmetric operad groups
• Virtual Thompson's group
• On Thompson Groups for Ważewski Dendrites
• Metric properties of higher-dimensional Thompson's groups
• Divergence functions of higher-dimensional Thompson's groups
• Towards computing the rational homology and assembly maps of generalised Thompson groups
• The power conjugacy problem in Higman-Thompson groups
• The word problem of the Brin-Thompson group is coNP-complete
• Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube
• A local treatment of finite alignment and path groupoids of nonfinitely aligned higher-rank graphs
• Distortion element in the automorphism group of a full shift