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A picture is better to explain what this means. And probably best explained concretely using, e.g., rotations and a few similar examples.
I don't quite understand it well enough to make the attempt, but I'll bet there's a mindblowing 5-minute explanation of what Gromov's theorem on groups of polynomial growth says.
Hmm. I'll have to think about that. I certainly don't know enough to be able to see immediately how or why that's interesting, so I'd have to dig into it. One result I did use to impress some 16 year olds was that in a group of even order there is an element a with a^2 = 1.
That's in a completely different league, but it served to show them something that without group theory wasn't obvious, but with group theory was completely obvious. That hinted at the power of the topic, but took about 2 minutes from nothing.
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