Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L 2 and a constant c > 0 such that for all x,y ∈ G we have ||f(x) - f(y)|| 2 ≥ cd(x,y) α. It was previously known that the Hilbert compression exponent of the wreath product ℤ ∼ ℤ is between 2/3 and 3/4. Here we show that 2/3 is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Coarse geometry
- Geometric group theory
- Hilbert compression exponents
- Markov type