# Characterization of countable groups as groups with a left-invariant distance with finite balls

In their book "Metric geometry of locally compact groups", Yves Cornulier and Pierre de la Harpe gave, the following characterization of countable groups: $$G$$ is countable if and only if it has a left-invariant metric with finite balls. See Proposition 1.A.1. and Lemma 2.B.5 of https://arxiv.org/pdf/1403.3796.pdf ,

I can easily see that if a group admits a left invariant metric such that every ball of finite radius is finite, then $$G$$ is countable. But what about the converse, how can we obtain a metric on $$G$$ with the above properties if $$G$$ is countable?

I am interested in the case when $$G$$ is discrete, so the proof given in the book is unnecessarily too complicated for my purposes.

• You refer to the introduction of the book... The introduction is an introduction, but the book includes proofs.
– YCor
Oct 28, 2019 at 23:16
• I found what I am looking for in Lemma 2.B.5 page 28, but I was trying to get a discrete version( like the one below given by Mark Sapir) as I think using that lemma to find such metric is like using Svarv-Milnor Lemma to prove a subgroup of finite index of finitely generated group is finitely generated. Oct 29, 2019 at 1:06
• Comments are not for extended discussion; this conversation has been moved to chat. Oct 30, 2019 at 19:42
• I am rolling back to the previous revision, which reflects what was said in the last comment by Hussain Rashed. If Hussain disagrees with that revision, then he may clarify. Oct 31, 2019 at 1:30

I have not found a proof in the book. But the statement is not difficult. Assign to each element of a countable group $$G$$ a natural number $$0,1,2,...$$ (for every $$n$$ only finite number of elements $$g$$ are assigned the same numbers $$n(g)=n$$, $$n(1)=0$$ and 1 is the only element with number 0, $$n(g^{-1})=n(g)$$). Then define a norm $$|g|$$ of $$g$$ as the smallest sum $$n(g_1)+...+n(g_k)$$ such that $$g=g_1g_2...g_k$$ in $$G$$. Then define the distance $$dist(g,h)=|g^{-1}h|$$. It is left invariant and has finite balls.