# Is it true that $\leq$ is independence of generating set?

Let $G$ be a finitely generated group and let $S$ be a finite generating set. (For convenience, we always assume our generating set is symmetric, i.e. $s\in S$ implies that $s^{-1}\in S$.) We define the word length $||g||_S$ of an element $g\in G- \{e\}$, with respect to the generators $S$, by $||g||_S=\inf\{k\geq 1: g= s_k s_{k-1}\ldots s_1, s_i\in S, 1\leq i\leq k\}$.

For $g, h\in G$, $h\leq_S g$ if and only if $||h||_S\leq ||g||_S$.

Question: Let $S,S^{*}$ be finite generating sets for $G$. Is it true that if $g\leq_S h$, then $g\leq_{S^{*}}h$?

A relation $\geq$ directs a nonempty set $D$ if an only if following are satisfied:

1. $\leq$ is reflexive and transitive

2. For each $a,b \in D$, there is a $c \in D$ such that $c \geq a$ and $c \geq b$

The pair $(D, \geq)$ is called a directed set.

It can be seen that $(G, \leq_S)$ is a directed set.

• I edited the question a bit, I am not really seeing the point of the whole directed set stuff though. Maybe include why you brought it up. This could just even be some like "I was thinking about directed sets (the def) and noticed groups are directed sets wrt generating sets, and the above question came to mind". Although, if there is no particular reason, it may be best to remove it
– user29123
Commented Jul 19, 2017 at 3:19

It is not true that word length comparisons are preserved by different generating sets. Just add whatever element you are comparing to a generating set, to make it length $1$.
Another simple, concrete example, consider the intergers $\mathbb Z$ under addition, and the standard generating set is $\{1, -1\}$, and you could also consider the generating set $\{\pm 2, \pm 3\}$, in one $2$ is longer than $1$ and in the other $1$ is longer than $2$. You can get more involved with different examples, like for non-abelian free groups you can consider different basis, and compare lengths between them.
There are restrictions though on how the different $\leq_X$ can be though, and in some sense this restriction is a starting point for the field geometric group theory. It ends up being that the "large scale geometry" does not change in a finitely generated group with different finite generating sets, and this is encoded by the concept of quasi-isometry. If $S,T$ are finite generating sets then it turns out that there is a constant $C$ such that $$\frac{1}{C} d_S(x,y)-C \leq d_T(x,y) \leq C d_S(x,y)+C,$$ where $d_X(x,y)$ is the length of $xy^{-1}$ with respect to the generating set $X$ (this can be thought of as the distance in the Cayley graph too). Drawing a more direct connection to your question, this means that $$\frac{1}{C} ||g||_S-C \leq ||g||_T \leq C ||g||_S+C.$$ Note that $C$ does depend on $S$ and $T$. Basically this says the global geometry is not that different.