How free is free ?
·1 min
Table of Contents
Automorphism, Geometry and the Growth of Groups #
Quasi isometry
Group generators induce metric
quasi isometry invariant properties: growth (focus) dehn fn virtual freeness (focus) hyperbolicity …
free group Nielsen rank formula Cayley graph metric
Ex: \(\left(\mathbb{Z}, \langle1\rangle\right) \sim_{\text{QI}} \left(\mathbb{Z}, \langle2,3\rangle\right)\)
Growth:
- Exponential
- intermediate (Grigorchuk; open problem: finitely presented but intermediate growth?)
- polynomial (Gromov \(\implies\) nilpotent)
Milnor conjecture (disproved by Grigorchuk) Milnor-Schwarz Thm
$$ \mathbb{R/Z} \sim S^1 $$\(\mathbb{R}^n\) is hyperbolic iff \(n = 1\) (Gromov thin hyperbolicity)
Nielsen transformation \(\text{Aut}(F_n)\)
GGT Martin Bridson Bible of GGT Gromov Hyperbolicity