Rank of a profinite group

Question

in the book Field Arithmetic by Fried and Jarden, two different definitions are given for the rank of a profinite group. The first, in chapter 16, is for finitely generated profinite groups, and this is the definition I am familiar with - namely, the rank of $G$ is the minimal cardinality of a generating set for $G$.

However, in section 17, there is another definition given for the rank, this time for nonfinitely generated profinite groups. First they define that a subset $X$ of a group $G$ converges to $1$ if for every open normal subgroup $N$ of $G$, $X \backslash N$ is a finite set. Then the rank of $G$ is defined as the cardinality of any set of generators that converges to $1$ (and they prove that such a set exists and the cardinality is always the same for a nonfinitely generated profinite group).

These two definitions do not coincide, as is mentioned in the book. However, they also mention that if you define the rank as "The minimal cardinality of a generating set that converges to $1$" then they do coincide. So I have a few questions:

1. Is this definition of the rank the standard one for topological groups, or is it special for profinite groups? In my reference books for topological groups I couldn't find this definition.

2. Why do they have two separate definitions for the rank instead of combining them into one definition?

3. Why is this condition about converging to $1$ necessary?

Answer 1

I spoke to my thesis supervisor on this question, and she sent me this reply:

"I gave a look at the definition of rank of a profinite group (Field arithmetic version) and I have some remarks for you:

1. This definition of rank is not standard even for profinite groups. In the literature, most people use the definition of p. 42 of Dixon et al. Perhaps you should add a remark about that in your thesis, because people that are used to the standard rank might be confused. It might be instructive to point out that those definitions are not equivalent.

2. I think the reason they divide the definition in the finitely and infinitely generated cases is just because they are constructing the rank in infinitely generated case. They state what is the rank, then they show that the definition they gave make sense, i.e. there is always a system of generators of G converging to 1, and two such systems have same cardinality. At the end, they remark that we can reformulate this definition so that it is valid for both cases. The way they present this definition is a bit confusing because at first glance it seems that the finite and infinite generated case are defined differently, but I really think it was just a (strange) way of presenting things...

3. As I said earlier, the condition of converging to 1 is necessary so that the rank is well defined. They could have defined a different rank in terms of cardinalities of sets with certain properties, as long as all sets with the chosen properties all have same cardinality.

4. We see that the usual rank of a profinite group is not the same as the rank in the definition of the Field Arithmetic. In particular, the usual rank is either finite or infinite, whereas the one of the Field Arithmetic is a cardinal number. I guess it makes sense that Helbig and the book use this definition, since both discuss S-rank, which is a cardinal as well...

If something is not clear, please let me know!"