What's an example of a group with equivalent uniform structures where multiplication is not uniformly continuous?

Question

Say we have a topological group $G$. It's easy to see that if $\cdot: G \times G \rightarrow G$ is uniformly continuous (with respect to either the right or left uniformity), then $G$ must have equivalent uniform structures. I figure the converse is probably false, on the basis that otherwise I would have seen this mentioned somewhere. But I can't think of an example, because I don't know many examples of topological groups, and all the examples I can think of with equivalent uniformities are built out of compact, abelian, or discrete groups, all of which do have uniformly continuous multiplication! Can anyone give me a counterexample? Or are these indeed equivalent?

Answer 1

Disclaimer: I haven't done the exercise carefully myself (and I'm not really in the mood to), but on Harry's request I'm posting it as an answer.

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According to Exercise 1.8.c on page 79 of Arhangel'skii-Tkachenko, Topological groups and related structures, the following are equivalent for a topological group (uniformly continuous means uniformly continuous with respect to both the left and the right uniform structures):

1. The multiplication map $G^{2} \to G$ is uniformly continuous.
2. The multiplication map $G^{n} \to G$ is uniformly continuous for all $n \geq 2$.
3. The group is balanced in the sense that the left and right uniformities are equivalent.

Since you're interested in $3 \implies 1$ that's good enough (provided that it is true).