## Distortion of finitely presented subgroups of non-positively curved groups I

We have seen that the Rips construction can be used to construct hyperbolic groups with finitely generated subgroups of enormous distortion — that is, not bounded above by any recursive function. However, those examples were not finitely presentable. This post and the next two will look at distortion of subgroups in non–positively curved groups when the subgroup can be finitely presented. [In fact, in almost all the examples we will examine, the heavily distorted subgroup will be a finite-rank free group.]

First a reminder —

What is distortion?

Definition. The distortion ${\textup{Dist}^{\Gamma}_H : {\mathbb N} \rightarrow {\mathbb N}}$ of a finitely generated subgroup ${H}$ in a finitely generated group ${\Gamma}$, is defined by

$\displaystyle \textup{Dist}^{\Gamma}_H(n) \ = \ \max \{ \, d_H(1,h) \, \mid \, h \in H, \ d_{\Gamma}(1,h) \leq n \, \}.$

Here ${d_{\Gamma}}$ and ${d_H}$ denote word metrics associated to some choices of finite generating sets. Those choices are not important: different choices lead to ${\simeq}$–equivalent distortion functions.

In this post we will give a number of examples culminating in some examples of Barnard, N. Brady, and P. Dani of CAT(-1) groups with free subgroups whose distortion outgrows any finite tower of exponentials.

Abelian subgroups of semi–hyperbolic groups

As we have seen, finitely generated abelian subgroups of semi–hyperbolic (for example, hyperbolic or CAT(0)) groups are quasi–isometrically embedded and so are undistorted — that is, their distortion functions grow like ${n \mapsto n}$ (in the sense of ${\simeq}$). This is the Algebraic Flat Torus Theorem.

Iterated Baumslag–Solitar groups

These examples are not non–positively curved, but they pave the way for non–positively curved examples to come. In each case, the subgroup we identify as having large distortion is ${{\mathbb Z}}$ and this leads to the Dehn functions being at least as fast growing as the distortion functions: if a short word ${w}$ represents a large power ${a^k}$ in this ${{\mathbb Z} = \langle a \rangle}$, then ${[a,w]}$ represents the identity and has area at least ${k}$. [It is not always the case that heavily distorted ${{\mathbb Z}}$ subgroups lead to large Dehn functions, but it is so for the examples that follow.] And, as we’ve previously discussed, in a group that has any right to be called non–positively curved, the Dehn function can grow no faster than quadratically.

The starting point for this family of examples is often referred to (inappropriately, as Gilbert Baumslag is known to point out!) as the Baumslag–Solitar group:

$\displaystyle \langle a, s \mid s^{-1} a s \ = \ a^2 \rangle.$

The subgroup ${{\mathbb Z} = \langle a \rangle}$ is exponentially distorted: the lower bound is apparent from the relation

$\displaystyle s^{-n} a s^n = a^{2^n}$

that stems from the doubling effect of ${s}$ when it conjugates ${a}$.
The figure below, which is a Van Kampan diagram for ${a^{2^n}s^{-n}a^{-1}s^n}$, graphically displays this calculation.

Suppose we now distort ${s}$ by introducing a new letter ${u}$ acting by ${u^{-1} s u = s^2}$. Then the distortion of ${{\mathbb Z} = \langle a \rangle}$ inside

$\displaystyle \langle a, s, u \mid s^{-1} a s =a^2, \ u^{-1} s u =s^2 \rangle$

grows like ${n \mapsto \exp^{(2)}(n)}$. [We will write ${\exp^{(l)}}$ for the ${l}$-fold iterate of the exponential function.] This time the lower bound comes from

$\displaystyle (u^{-n} s u^n)^{-1} a (u^{-n} s u^n) \ = \ s^{-2^n} a s^{2^n} \ = \ a^{2^{2^n}}.$

Iterating, we find the distortion of ${{\mathbb Z} = \langle a \rangle}$ inside

$\displaystyle \langle a, s_1, \cdots, s_l \mid {s_1}^{-1} a {s_1} =a^2, \ {s_{i+1}}^{-1} s_i s_{i+1} ={s_i}^2 \ (i >1) \rangle$

is ${\simeq \exp^{(l)}}$. [We will not go into details of the upper bounds, it suffices to say they do not harbour any great subtleties.] Here is an illustration of the calculation that leads to the 3–fold iterated exponential distortion:

This family has a limit (loosely speaking) — a one–relator group first described by Baumslag, and since analyzed by Gersten, Bernasconi (University of Utah thesis, 1994) and Platonov (Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2004, , no. 3, pp. 12–17; trans. in Moscow University Mathematics Bulletin, vol. 59 (2004), no. 3, pp. 12–17 (2005)) —

$\displaystyle \langle a, t \mid (t^{-1} a t)^{-1} a (t^{-1} a t) \ = \ a^2 \rangle.$

Introducing ${s}$ as shorthand for ${t^{-1} a t}$, we can re–express this presentation as:

$\displaystyle \langle a, s, t \mid s^{-1} a s = a^2, \ s = t^{-1} a t \rangle,$

and we see that conjugation by ${s}$ again has a doubling effect on ${a}$, and that ${t}$ conjugates ${a}$ to ${s}$ leads to a feedback effect whereby we get huge distortion of ${{\mathbb Z} = \langle a \rangle}$ on account of diagrams of the form shown below. To be precise, if the portion of the perimeter of this diagram that excludes the huge power of ${a}$ is to have length ${n}$, then the tree dual to the picture must have depth ${\simeq \lfloor \log_2 n \rfloor}$ and so the distortion grows like ${n \mapsto \exp^{(\lfloor \log_2 n \rfloor)}(1)}$.

Hyperbolic free–by-cyclic and free–by–free groups

Perhaps, on account of the connection with 3–manifolds that are mapping tori of surface diffeomorphisms, the natural first examples of non–positively curved groups with heavily distorted subgroups are free–by–cyclic and free–by–free groups where, in each case, the automorphism giving the action on the fibre distorts it exponentially.

A CAT(0) example of such a free–by–cyclic group is relatively easy to come by:

$\displaystyle F(a,b) \rtimes {\mathbb Z}$

where the generator ${t}$ for the ${{\mathbb Z}}$ factor acts by ${t^{-1} a t = ab}$ and ${t^{-1} b t = a}$. This is an exponentially growing automorphism — that is, ${t^{-n} a t^n}$ has length ${\simeq \exp(n)}$ when re–expressed as a minimal length word in ${a}$ and ${b}$. (In fact, the length grows exactly like the Fibonacci numbers.) This group can be presented as

$\displaystyle \langle a, b, t \mid t^{-1} a t = ab, \ t^{-1} b t = a \rangle,$

or equivalently as

$\displaystyle \langle a, b, c, t \mid at=cb, \ c=bt, \ c=ta \rangle.$

If we metrize the presentation 2–complex so that the cell corresponding to ${at=cb}$ is a Euclidean square and those corresponding to ${c=bt}$ and ${ c=ta}$ are Euclidean equilateral triangles, all of side–length ${1}$, then the link condition can be verified: one checks that the links “are large” — that is, all simple loops in the graph shown below have length at least ${2\pi}$. So the universal cover is a CAT(0)–space and the group is CAT(0).

Hyperbolic free–by–cyclic examples are a little harder to pin down. [For one thing the free group has to have rank at least three: Nielsen showed that any automorphism ${F(a,b) \rightarrow F(a,b)}$ takes ${a^{-1}b^{-1}ab}$ to a conjugate of itself or its inverse. It follows that ${F(a,b) \rtimes_{\phi} {\mathbb Z}}$ has a ${{\mathbb Z}^2}$–subgroup and so is not hyperbolic.] The existence of hyperbolic free–by–cyclic and, more generally, free–by–free examples can be derived from Bestvina–Feighn and Bestvina–Feighn–Handel. There are explicit CAT(-1) free–by–cylic examples in McCammond–Wise and CAT(-1) free–by–free examples (where the free group acting on the free fibre can have any finite rank) in N. Brady–A. Miller. All necessarily have exponentially distorted free fibres.

We will describe groups with free subgroups that are distorted like the iterated–exponential ${n \mapsto \exp^{(l)}(n)}$ or, faster, like ${n \mapsto \exp^{(\lfloor \log_4 n \rfloor)}(1)}$. These examples combine the techniques we’ve seen above for free–by–cyclic and the iterated Baumslag–Solitar groups. The idea of uniting these ideas to give hyperbolic examples is due to Mitra. We will describe subsequent examples of Barnard, Brady and Dani that are more explicit and are also CAT(-1).

Barnard, Brady and Dani use some of the combinatorial and geometric techniques created by Wise in his work on the Rips construction. As building blocks for their groups they use

$\displaystyle G_n \ = \ \langle \, a_1, \ldots, a_m, t_1, \ldots, t_n \mid {t_i}^{-1} a_j t_i = W_{ij} \ (1 \leq i \leq n, \ 1 \leq j \leq m) \ \rangle$

where ${m =14n}$ and the ${W_{ij}}$ are positive words on ${a_1, \ldots a_m}$ of length ${14}$ with the property that each two letter–word ${a_k a_l}$ appears at most once as a subword of the ${W_{ij}}$. They get such ${W_{ij}}$ explicitly by chopping up Wise’s word

$\displaystyle (a_1 a_1a_2 a_1 a_3 \cdots a_1 a_m) (a_2 a_2a_3 a_2 a_4 \cdots a_2 a_m) \cdots (a_{m-1} a_{m-1} a_m) a_m$

which has length ${m^2}$, and in which no ${a_ka_l}$ appears twice as a subword. In the same manner as the free–by–cyclic example described earlier they get:

Lemma. The subgroup ${\langle a_1, \ldots, a_m \rangle}$ is free of rank–${m}$ and is exponentially distorted in ${G_n}$.

Consider metrizing the presentation ${2}$–complex so that each cell is built from five right–angled hyperbolic pentagons:

One can check that the large–link condition is satisfied and so the complex is locally CAT(-1). So ${G_n}$ is CAT(-1). In fact, in the link the distance between distinct ${t_i^{\pm}}$ and ${t_j^{\pm}}$ is always at least ${2 \pi}$. Accordingly, Barnard–Brady–Dani describe the free subgroup generated by the ${t_i}$ as highly convex — this will be important momentarily.

Next, pursuing an analogy with the ${l}$-fold iterated Baumslag–Solitar examples, let

$\displaystyle H_l \ := \ G_1 \ast_{F_1} G_{14} \ast_{F_2} \ast G_{14^2} \ast \cdots \ast_{F_{l-1}} G_{14^{l-1}},$

where ${F_k}$ is a free group of rank ${14^k}$ which is identified with the exponentially distorted subgroup in ${G_{14^{k-1}}}$ and the highly convex subgroup in ${G_{14^{k}}}$. So we have exponential–upon–exponential–upon–exponential… distortion leading to ${G_{14^{l-1}}}$ being distorted ${\simeq \exp^{(l)}}$ in ${H_l}$. One can build a presentation 2–complex for a presentation of ${H_l}$ by gluing together copies of the presentation ${2}$–complexes we defined above for ${G_1}$, ${G_{14}}$, …, ${G_{14^{l-1}}}$. The key point is that the ${t_i}$‘s subgroups being highly convex prevents the link condition from failing. In summary:

Theorem. ${H_l}$ is a CAT(-1) group and the free group generated by the ${a_i}$‘s in the factor ${G_{14^{l-1}}}$ is distorted ${\simeq \exp^{(l)}}$ in ${H_l}$.

And, pursing an analogy with ${\langle a, s, t \mid s^{-1} a s = a^2, \ s = t^{-1} a t \rangle}$, define

$\displaystyle G \ := \ \langle \, a_1, \ldots, a_m, t_1, \ldots, t_n, s \mid {t_i}^{-1} a_j t_i = W_{ij}, \ s^{-1} a_k s = W_k \ \rangle$

where the ${W_{ij}}$ are ${mn}$ positive words of length ${14}$, the ${W_k}$ are ${m}$ positive words of length ${1}$, and there is no ${a_pa_q}$ subword occurring twice among any of them. Again, find these ${W_{ij}}$ and ${W_k}$ by carving up Wise’s word (${n=14^2}$ and ${m=14^3}$ will work).

Similarly to before, a locally CAT(-1) presentation 2–complex can be constructed out of 2–cells each of which is built out of five hyperbolic pentagons and the result is —

Theorem. ${G}$ is a CAT(-1) group and the free subgroup generated by the ${a_i}$ is distorted at least like ${n \mapsto \exp^{( \lfloor \log_4 n \rfloor )}(1)}$.

Non–positively curved examples of yet more extreme (Ackermannian) distortion will be described in the next post.