 v \text{ if either }|w|<|v|, \text{ or }|w|=|v|\text{ and }w\preceq' v.$$

For an element $g\in G$, by $W_g$ we denote the set $\{w\in W\colon w_G=g\}$ and by $w_g$ the minimal element of the set $W_g$, which is easily verified to exist, in $\preceq$. If there is no danger of confusion, for an element $g\in G$ we shall denote by $|g|$ the number $|w_g|$ which is equal to $d_S(g,1_G)$.

Finally, we use the linear order $\preceq$ on $W$ to define a linear order $\leq$ on $G$. For $g,h\in G$ we set $$g\leq h\text{ if }w_g\preceq w_h.$$

We call $G$ *shortlex combable*, with respect to a fixed symmetric generating set $S$ and a linear order on $S$, if there exists a constant $K\geq 1$ such that for every $g,h\in G$ with $d_S(g,h)=1$ and for every $i\leq \min\{d_S(g,1_G),d_S(h,1_G)\}$ we have $$d_S((w_g(\leq i))_G, (w_h(\leq i))_G)\leq K.$$

The constant $K$ will be called the *combability constant* of $G$.

First we show how such groups are useful for our purposes. Then we provide examples and show some applications. The following is the main result of this section.

\[thm:shortlex\] Let $G$ be a finitely generated shortlex combable group (with respect to $S$ equipped with some linear order). Then ${\mathcal{F}}(G,d_S)$ has a Schauder basis.

Since the linear order $\leq$ on $G$ is clearly isomorphic to the standard order on ${\mathbb{N}}$, we can use it to enumerate $G$ as $(g_n)_{n\in{\mathbb{N}}}$. For each $n\in{\mathbb{N}}$, set $G_n:=\{g_i\colon i\leq n\}$. For each $n$ we now define maps $P_n: G\rightarrow G_n$ as follows. First, set $m=\max\{|h|\colon h\in G_n\}$, then for $g\in G$, set $$P_n(g):=\begin{cases} g & \text{if }g\in G_n\\
(w_g(\leq m))_G & \text{if }g\notin G_n, (w_g(\leq m))_G\in G_n\\
(w_g(\leq m-1))_G & \text{otherwise}.
\end{cases}$$ We leave to the reader the straightforward verification that $P_n$ is well defined.\
[**Claim.**]{} The maps $(P_n)_{n\in{\mathbb{N}}}$ are uniformly bounded Lipschitz commutting retractions.\
First we check that for every $n,m\in{\mathbb{N}}$, $P_n\circ P_m=P_m