Super-exponential distortion of subgroups of CAT(−1) groups

Abstract

We construct 2-dimensional CAT(-1) groups which contain free subgroups with arbitrary iterated exponential distortion, and with distortion higher than any iterated exponential.

Hydra groups

“…This distortion of a free subgroup of a CAT(0) group stands in stark contrast to that of any abelian subgroup -they are always quasi-isometrically embedded (see Theorem 4.10 of Chapter III.Γ in [10], for example) and so no more than linearly distorted. The distortion we achieve exceeds that found in the hyperbolic groups of Mitra [29] and the subsequent 2-dimensional CAT(−1) groups of Barnard, Brady and Dani [2]: first, for all k, they give examples with a free subgroup of distortion ≃ exp (k) (n), and then they give examples with free subgroups whose distortion functions grow faster than exp (k) (n) for every k. However, our examples contain Z 2 subgroups and so are not hyperbolic.…”

“…Proof of Proposition 3.3. We prove (i) by induction on n. The inequality certainly holds for n = 2 since, by (6), (2). Now let n ′ > 2 and suppose that (i) holds for n < n ′ .…”

Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra

“…On the other hand, using Lemma 2.2(3) for the first inequality, Writing α := k 1 /s 8δ Γ +max{d Γ (e ,[î p i ,î p j ] Γ )|1≤i < j ≤3} , where k 1 is as per (2) in Section 2 applied to ∂ Γ , we then get d ∂ Γ (î (h n p i ),î (h n p j )) ≥ k 1 s −(î (h n p i ),î (h n p j )) Γ e ≥ α s n .…”

Hyperbolic hydra

“…Background. Other heavily distorted free subgroups of hyperbolic groups have been exhibited by Mitra [12]: for all k, he gives an example with a free subgroup of distortion like a k-fold iterated exponential function and, more extreme, an example where the number of iterations grows like log n. Barnard, the first author, and Dani developed Mitra's constructions into more explicit examples that are also CAT(−1) [3]. We are not aware of any example of a hyperbolic group with a finite-rank free subgroup of distortion exceeding that of our examples.…”