Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 Left Relatively Convex Subgroups
- 2 Groups with Context-free Co-word Problem and Embeddings into Thompson’s Group V
- 3 Limit Sets for Modules over Groups Acting on a CAT(0) Space
- 4 Ideal Structure of the C∗-algebra of R. Thompson’s group T
- 5 Local Similarity Groups with Context-free Co-word Problem
- 6 Compacta with Shapes of Finite Complexes: a Direct Approach to the Edwards–Geoghegan–Wall
- 7 The Horofunction Boundary of the Lamplighter Group L2 with the Diestel–Leader metric
- 8 Intrinsic Geometry of a Euclidean Simplex
- 9 Hyperbolic Dimension and Decomposition Complexity
- 10 Some Remarks on the Covering Groups of a Topological Group
- 11 The Σ-invariants of Thompson’s group F via Morse Theory
7 - The Horofunction Boundary of the Lamplighter Group L2 with the Diestel–Leader metric
Published online by Cambridge University Press: 27 August 2018
- Frontmatter
- Contents
- Contributors
- Preface
- 1 Left Relatively Convex Subgroups
- 2 Groups with Context-free Co-word Problem and Embeddings into Thompson’s Group V
- 3 Limit Sets for Modules over Groups Acting on a CAT(0) Space
- 4 Ideal Structure of the C∗-algebra of R. Thompson’s group T
- 5 Local Similarity Groups with Context-free Co-word Problem
- 6 Compacta with Shapes of Finite Complexes: a Direct Approach to the Edwards–Geoghegan–Wall
- 7 The Horofunction Boundary of the Lamplighter Group L2 with the Diestel–Leader metric
- 8 Intrinsic Geometry of a Euclidean Simplex
- 9 Hyperbolic Dimension and Decomposition Complexity
- 10 Some Remarks on the Covering Groups of a Topological Group
- 11 The Σ-invariants of Thompson’s group F via Morse Theory
Summary
We fully describe the horofunction boundary δhL2 with the word metric associated with the generating set {t, at} (i.e. the metric arising in the Diestel–Leader graph DL(2, 2)). The visual boundary δ∞L2 with this metric is a subset of δhL2. Although δ∞L2 does not embed continuously in δhL2, it naturally splits into two subspaces, each of which is a punctured Cantor set and does embed continuously. The height function on DL(2, 2) provides a natural stratification of δhL2, in which countably many non-Busemann points interpolate between the two halves of δ∞L2. Furthermore, the height function and its negation are themselves non-Busemann horofunctions in δhL2 and are global fixed points of the action of L2.
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- Chapter
- Information
- Topological Methods in Group Theory , pp. 111 - 134Publisher: Cambridge University PressPrint publication year: 2018
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