= Equivalence of finite word metrics
Changing a finite <generating set of a group> changes individual distances, but it cannot introduce arbitrarily large relative distortion. For two finite symmetric <generating sets of a group> $S,T$ of the same <group>, set
$$
K=\max\bigl(\{1\}\cup\{|s|_T:s\in S\}\bigr),\qquad L=\max\bigl(\{1\}\cup\{|t|_S:t\in T\}\bigr).
$$
Each $S$-letter can be replaced by at most $K$ $T$-letters, and conversely each $T$-letter by at most $L$ $S$-letters. Applying these substitutions to shortest representatives of $g^{-1}h$ gives
$$
\frac1L d_S(g,h)\leq d_T(g,h)\leq Kd_S(g,h).
$$
Thus the <identity map> is a <bilipschitz equivalence> between the two <word metrics>, and hence a <quasi-isometry>. The constants depend on the chosen <generating sets of a group>.https://webusers.imj-prg.fr/~kathryn.mann/classes/GGRminicourse.pdf[Mann, Proposition 1.2: bilipschitz equivalence]{ref}
In the <Cayley graphs of the integers> example, $S=\{-1,1\}$ and $T=\{-2,-1,1,2\}$ give $K=1$ and $L=2$: a unit move costs one in either alphabet, and a move of size two costs two unit moves. This recovers $d_T\leq d_S\leq2d_T$. The diamond and square in <word metric balls in the integer lattice> obey the same comparison, because $\max\{|m|,|n|\}\leq|m|+|n|\leq2\max\{|m|,|n|\}$.
Finiteness cannot simply be dropped from this universal statement. For the additive <integers>, take the infinite symmetric <generating set of a group> $U=\mathbb Z\setminus\{0\}$. Every nonzero displacement is a single allowed move, so $d_U(a,b)=1$ whenever $a\ne b$, whereas $d_S(0,n)=|n|$ for $S=\{-1,1\}$. There is no finite $L$ with $d_S\leq Ld_U$: choosing an <integer> $n>L$ violates it. This is a counterexample to unrestricted equivalence, not a claim that every infinite <generating set of a group> fails; two such sets with uniformly bounded generator lengths can still satisfy the same substitution bounds.https://webusers.imj-prg.fr/~kathryn.mann/classes/GGRminicourse.pdf[Mann, remark after Proposition 1.2: bounded metric from allowing all elements]{ref}
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