A Course on Geometric Group Theory
Abstract
These notes are based on a series of lectures I gave at the Tokyo Institute of Technology from April to July 2005. They constituted a course entitled “An introduction to geometric group theory ” totalling about 20 hours. The audience consisted of fourth year students, graduate students as well as several staff members. I therefore tried to present a logically coherent introduction to the subject, tailored to the background of the students, as well as including a number of diversions into more sophisticated applications of these ideas. There are many statements left as exercises. I believe that those essential to the logical developments will be fairly routine. Those related to examples or diversions may be more challenging. The notes assume a basic knowledge of group theory, and metric and topological spaces. We describe some of the fundamental notions of geometric group theory, such as quasi-isometries, and aim for a basic overview of hyperbolic groups. We describe group presentations from first principles. We give an outline description of fundamental groups and covering spaces, sufficient to allow us to illustrate various results with more explicit examples. We also give a crash course on hyperbolic geometry. Again the presentation is
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Euclidean space R^n for n ≥ 2 is not δ-hyperbolic for any δ ≥ 0, while the Poincaré ball D^n is log(1+√2)-hyperbolic.
“For instance, the Euclidean space R n for n ≥ 2 is not δ -hyperbolic for any δ ≥ 0 , while the Poincar´ e ball D n is log(1+ √ 2) -hyperbolic.”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Bowditch, 2006
For any δ > 0, any δ-hyperbolic metric space and any set of points, there exists a finite weighted tree and an embedding f such that for all i, j, a distance condition holds.
“Theorem : For any δ > 0 , any δ -hyperbolic metric space ( X,d X ) and any set of points x 1 , ..., x n ∈ X , there exists a finite weighted tree ( T, d T ) and an embedding f : T → X such that for all i, j ,”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Bowditch, 2006
The exponential growth property enables hyperbolic spaces to embed any weighted tree while almost preserving their metric.
“This exponential growth property enables hyperbolic spaces to embed any weighted tree while almost preserving their metric 3 (Gromov, 1987; Bowditch, 2006; Sarkar, 2011).”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Bowditch, 2006
Hyperbolic spaces provably model tree-like structures better than Euclidean geometry.
“Following prior work, we first advocate for using hyperbolic spaces which provably model tree-like structures better than Euclidean geometry.”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Bowditch, 2006
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Hyperbolic Entailment Cones for Learning Hierarchical Embeddings
Ganea, O.-E., Bécigneul, G., & Hofmann, T. (2018) · arXiv · doi:10.48550/arxiv.1804.01882 · Cited by 115 · see citing works
- Authors
- Bowditch, B. H.
- Corresponding author
- Brian H. Bowditch
- Venue
- Mathematical Society of Japan Memoirs
- Publisher
- The Mathematical Society of Japan
- Published
- 2006
- ISSN
- 2189-1494
- DOI
- 10.1142/e003
- OpenAlex
- W654957503
- Cited by
- 40
- References
- 0
- Citations by year
- 2026: 2 · 2024: 4 · 2023: 2 · 2022: 1 · 2021: 3 · 2020: 2 · 2019: 3 · 2018: 2 · 2017: 6 · 2016: 3 · 2015: 2 · 2014: 2 · 2013: 3 · 2012: 1
- FWCI
- 0.41
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- WORLD SCIENTIFIC eBooks (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Mathematics › Geometry and Topology › Mathematics and Applications
- Keywords
- geometric group theory, quasi-isometry, hyperbolic groups, group presentations, fundamental group, covering spaces, hyperbolic geometry
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