Embeddings with Given Length Functions

Here we present the main ideas of the proofs of results from [30] (see Section 1.3).

If G is a subgroup of a group H with a finite set of generators $B=\{b_1,\dots,b_n\}$ then the function $\ell(g)=\vert g\vert _B$ on G evidently satisfies conditions (D1)-(D3) from Theorem 8. For instance condition (D3) holds because the number of all words of length at most k in the alphabet B grows exponentially as $k\rightarrow\infty$.

To prove that every function $G\to {\bf N}$ satisfying conditions (D1)-(D3) can be realized as the length function of G inside a finitely generated group H, we start with a presentation G = F_G/N, where F_G is a free group with the basis $\{x_g\}_{g\in G\backslash \{1\}}$ and N is the kernel of homomorphism $\epsilon: x_g\mapsto g$.

Then we construct an embedding $\beta: x_g\to X_g$ of F_G into the 2-generated free group F=F₂=F(b₁,b₂), such that the image $\beta(F_G)$is freely generated by X_g, $g \in G\backslash \{1\}$ and the words X_gare very ``independent" in the sense described below.

The group H is equal to the quotient F/L, where L is the normal closure of $\beta(N)$ in F.

Notice that whatever homomorphism $\beta$ we choose, it induces a homomorphism $\gamma: G\rightarrow H$. To make this homomorphism injective, we need the following property:

$$L\cap\beta(F_G) = \beta(N).$$

In fact $\beta(F_G)$ satisfies the following much stronger property:

(*) For any normal subgroup $U\triangleleft \beta(F_G)$ there is a normal subgroup $V\triangleleft F$ such that $U=V\cap \beta(F_G)$.

The fact that free groups and more generally every non-elementary hyperbolic group has plenty of infinitely generated free subgroups with property (*) is interesting in its own right, it was the key ingredient in Olshanskii's proof from [29] of the fact that every non-elementary hyperbolic group is SQ-universal.

It turns out that we can make $\beta$ satisfy condition (*) by choosing reduced words X_g with the following small cancellation condition:

(**) If Y is a subword of a word X_g and $\vert Y\vert\ge\frac{1}{50}\vert X_g\vert$then Y occurs in X_g as a subword only once, and Y occurs neither in X_g^-1 nor in $X_h^{\pm 1}$ for $h\ne g$.

It is relatively easy to construct an infinite set of words X_g in the alphabet $\{b_1,b_2\}$ which satisfies the (**)-condition and has exponential growth, that is the number of different words X_gof length k grows exponentially as $k\to \infty$.

Since by condition (D3) the number of elements $g\in G$ with $\ell(g)\le k$ does not exceed c^k for some constant c, we can choose the set $\{X_g\}$ in such a way that

$$\ell(g)\le \vert X_g\vert<d\ell(g)$$

for some positive constant dand every $g \in G\backslash \{1\}$.

We need to show that the embedding $\gamma: G\to H$ is quasi-isometric. For this we take any element X_g of $\gamma(G)$ and consider the shortest word W in the alphabet $\{b_1,b_2\}$representing X_g in H. The group H is given by the presentation consisting of all relations of the form X_g1X_g2...X_gn=1 where g₁g₂...g_n=1 in G.

Since X_g=W modulo this presentation, we can consider the corresponding van Kampen diagram $\Delta$with boundary label X_gW^-1. We can assume that the number of cells in $\Delta$ is minimal among all such diagrams.

The condition (**) implies the following property of van Kampen diagrams over the presentation of H. Let $\Pi_1$ and $\Pi_2$ be cells in a diagram $\Delta$having a common edge. Then either any common arc p of the boundaries $\partial\Pi_1$ and $\partial\Pi_2$ is short compared to the perimeters P₁, P₂ of the cells (say, $\vert p\vert\le\frac{1}{10}\min(P_1, P_2)$), or a subdiagram consisting of $\Pi_1$ and $\Pi_2$, has also a boundary label of the form $w=w(X_{g_1},\dots,X_{g_n})$, i.e. the subdiagram can be replaced by one cell. The latter option cannot occur in $\Delta$ because of the minimality of the diagram $\Delta$. Thus $\Delta$ satisfies a small cancellation condition [25].

This in turn allows us to prove that the word W is freely equal to a product $X_{g_1}^{\pm 1}\dots X_{g_s}^{\pm 1}$ for some $g_j\in G$ with $g=g_1^{\pm 1}\dots g_s^{\pm 1}$ (see [30] for details). Further, since the cancellations in such a product are small,

$$\vert\gamma(g)\vert _H\ge (1-\frac{2}{50})\sum_{j=1}^s\vert X_{g_1}\vert.$$

By conditions (D1), (D2), and by the choice of X_g, we have:

$$\vert\gamma(g)\vert _H\ge 0.96\sum \ell(g_j)=0.96\sum \ell(g_j^{\pm 1})\ge 0.96 \ell(g).$$

Hence $0.96 \ell(g)\le \vert\gamma(g)\vert _H\le d\ell(g)$, so $\ell$ is O-equivalent to the length function of G in H.