Do uncountable groups have generating sets where you can find elements of arbitrarily long length, and what are conditions to guarantee that? I would like to find a way to pick a set of generators in a group $G$ so that one can always find an element of $G$ of arbitrary length.  I'm not sure whether or not this is always possible, and if not, what conditions on $G$ make it possible?
 A: In general there are uncountable groups where for all generating sets, every element has length bounded by some $n$.


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*In Generating infinite symmetric groups by George M. Bergman, it is shown that $S_\Omega$, where $\Omega$ is an infinite set then if you look at the Cayley graph generated by a generating set $U$, that Cayley graph is bounded. He also shows that for $S_\Omega$ there is not a uniform bound for all generating sets.

*A paper by Yves de Cornulier, Strongly bounded groups and infinite powers of finite groups, shows that if $G$ is a finite perfect group and $I$ is a set, then $G^I$ has a bounded Cayley graph for every generating set.

*Shelah constructed such a group in On a problem of Kurosh, Jónsson groups, and applications too. In fact there is a uniform bound that can be choosen (unlike $S_\Omega$).


It seems like it may be open if there are countable groups with that property. (This is question 8 in Bergman's paper)
Prop 2.4 in Yves's paper says that these properties for groups are equivalent:


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*(not strongly bounded) There is a metric space it acts on with an unbounded orbit

*(not cayley bounded or has cofinality=$\omega$) There is an unbounded Cayley graph, or $G$ is the countable union of a strictly increasing chain of subgroups


That condition could be useful, if you know that your group has cofinality$\neq \omega$, and it acts on a metric space with unbounded orbits (so there must be an unbounded Cayley graph). There is another condition you could use, Prop 2.7, from the same paper.
Maybe something else that could be useful, say your group can be a topological group which is not compact, but it is compactly generated, then you have that the compact set gives an unbounded Cayley graph. Infinite finitely generated groups can be given the discrete topology and then the finite generating set will be the compact generating set, $(\mathbb{R}^n,+)$, with standard topology is generated by $[0,1]^n$.
