Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Tools and Theory
- Part II Results and Applications
- 5 Growth, Dimension, and Heat Kernel
- 6 Bounded Harmonic functions
- 7 Choquet–Deny Groups
- 8 The Milnor–Wolf Theorem
- 9 Gromov’s Theorem
- Appendices
- Appendix A Hilbert Space Background
- Appendix B Entropy
- Appendix C Coupling and Total Variation
- References
- Index
5 - Growth, Dimension, and Heat Kernel
from Part II - Results and Applications
Published online by Cambridge University Press: 16 May 2024
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Tools and Theory
- Part II Results and Applications
- 5 Growth, Dimension, and Heat Kernel
- 6 Bounded Harmonic functions
- 7 Choquet–Deny Groups
- 8 The Milnor–Wolf Theorem
- 9 Gromov’s Theorem
- Appendices
- Appendix A Hilbert Space Background
- Appendix B Entropy
- Appendix C Coupling and Total Variation
- References
- Index
Summary
In this chapter we start applying the tools developed in Part I to study random walks.The notion of amenable groups is defined, and Kesten’s criterion for amenable groups is proved. We then move to define the notion of isopermitric dimension. Inequalities relating the volume growth of a group to the isoperimetric dimension and to the decay of the heat kernel are proved.
Keywords
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- Information
- Harmonic Functions and Random Walks on Groups , pp. 161 - 195Publisher: Cambridge University PressPrint publication year: 2024