Exponentially distorted subgroups
in wreath products

The main result of this paper is –

Distortion of finitely generated subgroups in finitely generated groups is foundational and widely studied. It compares a subgroup’s word metric with the restriction of the word metric of the ambient group. For example, the subgroup $H=\langle a\rangle\cong\mathbb{Z}$ of $G=\langle a,t\mid t^{-1}at=a^{2}\rangle$ is said to be at least exponentially distorted because for all $n\in\mathbb{N}$, the length-$2^{n}$ word $a^{2^{n}}$ equals the length-$(2n+1)$ word $t^{-n}at^{n}$ in $G$; in fact, it is exponentially distorted because, moreover, there is a constant $C>0$ such that whenever a word on $a$ and $t$ of length $k$ represents an element $a^{L}$ of $H$, we have $L\leq C^{k}$. The Heisenberg group $G=\langle a,b,c\mid[a,c],[b,c],[a,b]c^{-1}\rangle$ provides another example: its center $\langle c\rangle\cong\mathbb{Z}$ is at least quadratically distorted because $[a^{n},b^{n}]=c^{n^{2}}$ in $G$ for all $n\in\mathbb{N}$; and, in fact, it is quadratically distorted because, moreover, there exists $C>0$ such that for every word on $a$, $b$, and $c$ of length $k$ that represents an element $c^{L}$ of $H$, we have $L\leq Ck^{2}$.

In some fundamental cases subgroup distortion is well-behaved. Subgroups of finitely generated free groups and of fundamental groups of closed hyperbolic surfaces are undistorted [Pit93, Sho91]. Subgroups of finitely generated nilpotent groups are all at most polynomially distorted [Osi01]. But subgroup distortion can be wild even in some seemingly benign groups. There are subgroups of $F_{2}\times F_{2}$ and of rank-3 free solvable groups whose distortion functions cannot be bounded from above by a recursive function [Mih66, Umi95].

Theorem 1.1 is a next step in a direction of inquiry pursued by Davis and Olshanskii [Dav11, DO11]. They proved that every subgroup of $\mathbb{Z}\wr\mathbb{Z}$ is distorted like $n^{d}$ for some positive integer $d$ and, for all such $d$, they exhibited a subgroup realizing that distortion. Davis [Dav11] suggested next exploring subgroup distortion in $\mathbb{Z}\wr F_{n}$ and quoted speculation that an answer would be of interest for the study of von Neumann algebras. Theorem 1.1 reveals a sharp contrast between subgroup distortion in $\mathbb{Z}\wr\mathbb{Z}$ and in $\mathbb{Z}\wr F_{2}$. Also, given that all finitely generated subgroups in $\mathbb{Z}$ and in $F_{2}$ are undistorted and in $\mathbb{Z}\wr\mathbb{Z}$ are at most polynomially distorted, it shows that the wreath-product construction can give rise to substantial subgroup distortion.

The most novel feature of the work here is the idea behind the exponential lower bound (as proved in Section 3). It relies on the observation that $F_{2}$ and $\mathbb{Z}\wr\mathbb{Z}$ admit height functions (homomorphisms onto $\mathbb{Z}$) such that for all integers $n>0$, there are pairs of height-$0$ elements a distance $2n$ apart with the property that any path from one to the other travels up to height $n$ en route—see Proposition 3.1.

An intrinsic description of the subgroup $H$ is not immediately evident from its definition in Theorem 1.1. Our proof of the exponential upper bound on its distortion in Section 4 includes such a description.

The second theorem of this article makes the point that our subgroups of Theorem 1.1 are necessarily delicate (given that all subgroups are undistorted in $\mathbb{Z}$ and in $F_{2}$) in that exponentially distorted subgroups have to intersect both factors of the wreath product non-trivially and cannot be $\mathbb{Z}$-subgroups. Closely related results can be found in [BLP15], which we recommend for a more detailed treatment than the proof we outline in Section 5.

In the case of $G=\mathbb{Z}\wr F_{2}$ all the subgroups in this list are undistorted in $G$, because all finitely generated subgroups of $F_{2}$ are undistorted. In the case of $G=\mathbb{Z}\wr(\mathbb{Z}\wr\mathbb{Z})$ the list includes polynomially distorted subgroups on account of [Dav11, DO11].

It is tempting to try to use the results in this paper to address the question of Guba & Sapir [GS99] as to what functions may be distortion functions of finitely generated subgroups of Thompson’s group $F$. However they do not speak to that because, while $(\mathbb{Z}\wr\mathbb{Z})\wr\mathbb{Z}$ is a subgroup of Thompson’s group, $\mathbb{Z}\wr(\mathbb{Z}\wr\mathbb{Z})$ and $\mathbb{Z}\wr F_{2}$ are not [Ble08, Theorem 1.2].

A companion to this paper is in preparation, proving that for $\mathbb{Z}\wr F_{2}$, Theorem 1.1 is best possible—that is, every finitely generated subgroup of $\mathbb{Z}\wr F_{2}$ is at most exponentially distorted [BR].

I am most grateful to Aria Beaupré, Jim Belk, Conan Gillis, and Chaitanya Tappu for helpful and stimulating conversations and I thank an anonymous referee for generously thoughtful feedback on the exposition, greatly enhancing this paper.

This paper is dedicated to the memory of Peter Neumann in tribute to his foundational work both as a scholar (e.g., pertinently, [Neu64]) and as a teacher.

For a group $G$ with finite generating set $S$, let $|g|_{S}$ denote the length of a shortest word on $S^{\pm 1}$ representing $g$. The word metric $d_{S}$ on $G$ is $d_{S}(g,h)=\left|g^{-1}h\right|_{S}$.

Suppose a subgroup $H\leq G$ is generated by a finite set $T\subseteq G$. The distortion function $\hbox{\rm Dist}^{G}_{H}:\mathbb{N}\to\mathbb{N}$ for $H$ in $G$ compares the word metric $d_{T}$ on $H$ to the restriction of $d_{S}$ to $H$:

For functions $f,g:\mathbb{N}\to\mathbb{N}$ we write $f\preceq g$ when there exists $C>0$ such that $f(n)\leq Cg(Cn+C)+Cn+C$ for all $n$. We write $f\simeq g$ when $f\preceq g$ and $g\preceq f$.

Two finite generating sets for a group yield biLipschitz word metrics, with the constants reflecting the minimal length words required to express the elements of one generating set as words on the other. So, up to $\simeq$, the growth rate of a distortion function does not depend on the finite generating sets.

Let $W=\bigoplus_{K}L$, the direct sum of a $K$-indexed family of copies of $L$. The (restricted) wreath product $G=L\wr K$ is the semi-direct product $W\rtimes K$ with $K$ acting to shift the indexing. More precisely, given a function $f:K\to L$ that is finitely supported (meaning $f(k)=e$ for all but finitely many $k\in K$) and given $k\in K$, define $f^{k}:K\to L$ by $f^{k}(v)=f(vk^{-1})$. Then $L\wr K$ is the set of such pairs $(f,k)$ with multiplication

A lamplighter description helps us navigate $G$. Suppose $\left\{a_{1},\ldots,a_{m}\right\}$ generates $L$ and $\left\{b_{1},\ldots,b_{l}\right\}$ generates $K$. Viewing $L$ and $K$ as subgroups of $G$, with $L$ being the $e$-summand of $W$, the set $S=\left\{a_{1},\ldots,a_{m},b_{1},\ldots,b_{l}\right\}$ generates $G$. Then $W$ is the normal closure of $a_{1},\ldots,a_{m}$ in $G$, or equivalently the kernel of the map $\Phi:G{\kern 3.0pt\to\kern-8.0pt\to\kern 3.0pt}K$ that kills $a_{1},\ldots,a_{m}$. Imagine $K$ as a city. At each street corner (that is, each element of $K$) there is a lamp whose setting is expressed as an element of $L$. An $(f,z)\in L\wr K$ records settings $f(k)\in L$ of the lamps $k\in K$ and a location $z\in K$ for the lamplighter. A word $w$ on $S^{\pm 1}$ representing $(f,z)$ describes how at dusk a lamplighter walks the city streets adjusting the lamps to achieve $(f,z)$. He starts at $e\in K$ with all lights off (that is, set to $e\in L$) and, reading $w$ from left to right, moves in $K$ according to the $b_{1}^{\pm 1},\ldots,b_{l}^{\pm 1}$ until finally arrives at $z$. En route, he adjusts the setting of each lamp where he stands according to the $a_{1}^{\pm 1},\ldots,a_{m}^{\pm 1}$.

Our conventions are that $[x,a]=x^{-1}a^{-1}xa$ and $x^{a}=a^{-1}xa$.

For example, when $K=F_{2}=F(x,y)$, because the Cayley graph is a tree, the proposition applies with $P_{n}$ the set of reduced words whose prefixes $\pi$ all satisfy $\theta(\pi)<n$.

We view (ii) as saying that when moving in the Cayley graph of $K$, it is not possible to enter $P_{n}$ from below, and (iii) as saying that $P_{n}$ can only be entered from above by moving from a height-$n$ element outside $P_{n}$ to a height-$(n-1)$ element in $P_{n}$. Together, (i) and (ii) imply that $x^{i}\in P_{n}$ if and only if $i<n$.

Here is the idea behind this proposition in terms of the lamplighter description.

Suppose the lights at the elements $e$ and $x^{n}y^{-n}$ of $K$ are set to $1$ and $-1$, respectively, and all other lights are off (set to $0$). How can a lamplighter turn all the lights off using $x$, $y$, $\sigma$, and $\tau$? He has four types of moves at his disposal: he can navigate the Cayley graph of $K$ (by using $x$ and $y$); because $\sigma=[x,a]a=x^{-1}a^{-1}xa^{2}$, he can decrement by $1$ the lamp one step away in the $x^{-1}$-direction at the expense of incrementing the lamp where he stands by $2$; and likewise in the $y^{-1}$-direction using $\tau$. The answer is he sets the lamp at $e$ to $0$ at the expense of setting the lamp at $x$ to $2$. Then he sets the lamp at $x$ to $0$ at the expense of setting the lamp at $x^{2}$ to $4$. And so on, until the lamp at $x^{n}$ is set to $2^{n}$. He then sets that to $0$ and, proceeding in the $y^{-1}$ direction, sets the lamp at $x^{n}y^{-1}$ to $2^{n-1}$. Continuing likewise in the $y^{-1}$-direction he sets the lamp at $x^{n}y^{-(n-1)}$ to $2$. Finally, he adjusts the lamp at $x^{n}y^{-(n-1)}$ to zero at the expense of changing the lamp at $x^{n}y^{-n}$, but as that was initially set to $-1$, this results in all lights being off, as required.

The above method takes at least $2^{n}$ moves, but could it have been accomplished with fewer? The hypothesis involving $P_{n}$, $x^{n}y^{-n}$, and the epimorphism $K\to\mathbb{Z}$ ensures it cannot. Any path from $e$ to $x^{n}y^{-n}$ in the Cayley graph must rise to height $n$ to escape $P_{n}$ and the settings of the lights must be incrementally adjusted on the way up so that the number of $\sigma$- and $\tau$-moves grows exponentially with the height.

Here is a proof.

(In fact, when $K$ is $F_{2}$ or $\mathbb{Z}\wr\mathbb{Z}$, as per the following corollary, the roles of $x$ and $y$ are interchangeable and the above proof shows $\lambda_{n}\mu_{n}^{-1}$ is a geodesic word.)

The same proof works for $\mathbb{Z}\wr(C\wr\mathbb{Z})$ for any finite cyclic group $C\neq\{1\}$.

An example where Proposition 3.1 does not apply may be illuminating. The hypotheses on $P_{n}$ imply that any path from $e$ to $x^{n}y^{-n}$ in the Cayley graph of $K$ must climb to height $n$ en route. If $K=\mathbb{Z}^{2}=\langle x,y\mid[x,y]\rangle$, then this need not happen, because $x^{n}y^{-n}=(xy^{-1})^{n}$. Indeed, in $\mathbb{Z}\wr\mathbb{Z}^{2}$ we find that $a^{-1}x^{n}y^{-n}a=\left(a^{-1}xy^{-1}a\right)^{n}=((x\sigma)(y\tau)^{-1})^{n}$, a word of length $4n$ on the generators of $H$.

Let $G=\mathbb{Z}\wr K$ where $K$ is $F_{2}$ or $\mathbb{Z}\wr\mathbb{Z}$ as per Theorem 1.1. Let $\theta:K\to\mathbb{Z}$ be the epimorphism mapping $x$ and $y$ to $1$.

The above argument is quantified in such a way that a couple of further observations complete the exponential upper bound proof for Theorem 1.1. The absolute values of the settings of the lamps along the at most $n$ paths will grow to at most $n+n2^{n}$ in the course of the transformation of the lamp settings. The number of times $x^{\pm 1}$ and $y^{\pm 1}$ are used (for the movement) is at most $n^{2}$. So $u$ has length at most a constant times $2^{n}$, establishing the exponential upper bound on $\hbox{\rm Dist}^{G}_{H}$.

Here we prove Theorem 1.2. We have $G=\mathbb{Z}\wr K$, where $K$ is a finitely generated group. So $G=W\rtimes K$, where $W=\bigoplus_{K}\mathbb{Z}$.

Claim (1) is that if $H$ is a finitely generated subgroup of $W$, then $H$ is undistorted in $G$. Well, because $H$ is finitely generated, it is a subgroup of the product of finitely many of the summands in $W=\bigoplus_{K}\mathbb{Z}$ and there exists $C\geq 1$ such that for all $g=(f,e)\in H$, both $d_{G}(e,g)$ and $d_{H}(e,g)$ (word metrics with respect to the generating sets for $G$ or for $H$, respectively) are between $\frac{1}{C}\max_{i\in K}|f(i)|$ and $C\max_{i\in K}|f(i)|$. So $H$ is undistorted in $G$.

Claim (2) is that if $H$ is a finitely generated subgroup of $K$, then $\hbox{\rm Dist}^{G}_{H}\simeq\hbox{\rm Dist}^{K}_{H}$. This is straight-forward on account of the map $G{\kern 3.0pt\to\kern-8.0pt\to\kern 3.0pt}K$ killing $W$.

Finally, claim (3) is that if $\hat{H}=\langle t\rangle$ is a $\mathbb{Z}$-subgroup of $G$, then either $\hat{H}$ is undistorted in $G$ or there exists a subgroup $H\cong\mathbb{Z}$ of $W$ or $K$ such that $\hbox{\rm Dist}^{G}_{H}\simeq\hbox{\rm Dist}^{G}_{\hat{H}}$.

Well, $t=(f,k)$ for some $f\in W$ and some $k\in K$. If $k$ has finite order $r$, then $t^{r}=(f^{\prime},e)$ for some $f^{\prime}\in W$, and $H=\langle t^{r}\rangle$ is a subgroup of $W$ such that $\hbox{\rm Dist}^{G}_{H}\simeq\hbox{\rm Dist}^{G}_{\hat{H}}$.

Suppose, on the other hand, $k$ has infinite order. Roughly speaking, we will show that for all $j$, either $t^{j}$ illuminates lights close to most of $e,k,\ldots,k^{j}$ and $\hat{H}$ is therefore undistorted in $G$, or it only illuminates lights close to $e$ and $k^{j}$, and $\hat{H}$ is therefore distorted in $G$ similarly to $\langle(\mathbf{0},k)\rangle$.

Let $F:K\to\mathbb{Z}$ be the map $\sum_{i\in\mathbb{Z}}f^{k^{i}}$. In terms of the lamplighter model, $F$ tells us the settings of the lights after the lamplighter acts per $f$ at $k^{i}$ for every $i\in\mathbb{Z}$. As $k$ has infinite order, $F$ is well-defined—for any $h\in K$, $f^{k^{i}}(h)=0$ for all but finitely many $i$—but $F$ need not be finitely supported and so may not represent an element of $W$. Indeed, $F$ is invariant under the action of $k$, so either $F=\mathbf{0}$ (the zero-map), or $F$ has infinite support.

For $j\geq 1$, let $f_{j}=\sum_{i=0}^{j-1}f^{k^{i}}$, so that $t^{j}=(f_{j},k^{j})$.

Let $L>0$ be sufficiently large that $\hbox{\rm supp}f\subset N_{L}(e)$—that is, the radius-$L$ neighbourhood of $e$ in the Cayley graph of $K$ contains the support of $f$. Then $\hbox{\rm supp}F\subseteq N_{L}(\langle k\rangle)$ and $\hbox{\rm supp}f_{j}\subseteq N_{L}(\{k^{0},k^{1},\ldots,k^{j}\})$ for all $j\geq 1$.

Because $k$ has infinite order, for all $R>0$, there exists $i$ such that $k^{i},k^{i+1},\ldots$ are a distance greater than $R$ from $e$ in the Cayley graph of $K$. It follows that there exists $C>0$ such that for all $j>0$, the functions $F$ and $f_{j}$ agree on $\mathcal{N}_{j}:=N_{L}(\{k^{C},k^{C+1},\ldots,k^{j-C}\})$. So, as $F$ is $k$-invariant, $f_{j}$ follows the same repeating pattern as $F$ along $\mathcal{N}_{j}$—more precisely, the restrictions of $f_{j}$ to $N_{L}(k^{C})$, to $N_{L}(k^{C+1})$, …, and to $N_{L}(k^{j-C})$ all agree after translations by successive powers of $k$. And therefore, if $\hbox{\rm supp}F\neq\emptyset$, there exists $\lambda,\mu>0$ such that for all $j>0$ we have $d_{G}(e,t^{j})\geq\lambda j-\mu$, because to achieve the element $t^{j}\in G$, the lamplighter must visit every one of these neighbourhoods. So $\hat{H}$ is undistorted in $G$. And if, on the other hand, $\hbox{\rm supp}F=\emptyset$, then there exists $\nu>0$ such that for all $j$ and all $g\in K\smallsetminus N_{\nu}(\{e,k^{j}\})$, we have $f_{j}(g)=e$. So $d_{G}(e,t^{j})\leq d_{G}(e,u^{j})+C$ where $u=(\mathbf{0},k)$. So $\hbox{\rm Dist}^{G}_{H}\simeq\hbox{\rm Dist}^{G}_{\hat{H}}$ where $H:=\langle u\rangle$, which is a subgroup of $K$.