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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.

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    $\begingroup$ You refer to the introduction of the book... The introduction is an introduction, but the book includes proofs. $\endgroup$
    – YCor
    Oct 28, 2019 at 23:16
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    $\begingroup$ 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. $\endgroup$ Oct 29, 2019 at 1:06
  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Todd Trimble
    Oct 30, 2019 at 19:42
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    $\begingroup$ 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. $\endgroup$
    – Todd Trimble
    Oct 31, 2019 at 1:30

1 Answer 1

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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.

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  • $\begingroup$ +1 Thank you very much, Prof. Mark Sapir! $\endgroup$ Oct 29, 2019 at 1:59

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