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
9 - Gromov’s Theorem
from Part II - Results and Applications
Published online by Cambridge University Press: 16 May 2024
- 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
This chapter provides a full elementary proof of Gromov’s theorem, which states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. The proof proceeds along the ideas laid forth by Ozawa, using the existence of a harmonic cocycle. Gromov’s theorem is then used to classify all recurrent groups. Also, consequences of harmonic cocycle to diffusivitiy of the walk are shown.
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- Harmonic Functions and Random Walks on Groups , pp. 301 - 354Publisher: Cambridge University PressPrint publication year: 2024