> What prevents a group from having a "pointy tail", that is,
> a "corridor" of elements at increasing distances from the
> identity? In fact, does the number of elements as a
> function of distance have to be unimodal? Could this
> function have a "waist"? Intuitively, this sounds
> impossible, but I am wondering what constraints on such
> functions are known.
>
It seems to me it can't be too pointy. Working backwards, the
number of arrangements working from the end has to explode
(probably in symetry) with the number of arrangements form the
start.