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Concept

Growth rate

How fast the count of elements within a given distance of the identity rises in a group. It survives a change of generating set up to rescaling the distance, so polynomial and exponential growth are properties of the group, and polynomial growth forces a nilpotent structure.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

p(n) to 60, against the Hardy–Ramanujan estimate. The number of partitions of each number up to sixty on a logarithmic scale, with the asymptotic estimate drawn over it and the ratio of the two tabulated.

The size of a number with no formula

There is no closed expression for the number of partitions of n. There is an expression for how large it is — with a square root in the exponent and a π in front — and it is accurate enough that rounding a few terms of its refinement gives the exact count.

number · Partitions The ball around the identity, in 3 groups. A table of the number of group elements within each distance of the identity, one row per group, with the growth type each row exhibits beside it.

How fast the ball fills

Count the elements within r steps of doing nothing. The count grows like a polynomial in some groups and like a power of three in others, the distinction survives every change of generating set, and which polynomial degrees are possible is a theorem nobody expected.

algebra · Cayley graph The share of a ball that is its own edge. A plot of the proportion of each ball formed by its outermost shell against the radius, one line per group — falling towards nothing for the lattice groups and holding steady for the free group.

The edge that is as big as the ball

In a lattice the boundary of a large ball is a negligible fraction of it. In a tree it is two thirds of it at every size — and that single ratio, not the group's size, is what decides whether a set can be cut into pieces and reassembled into two copies of itself.

algebra · Cayley graph

Named alongside it

The objects these essays reach for when they reach for this one.

Cayley graphFree groupGroup actionInvariantWord metricAnalytic continuationApproximationAsymptoticsAxiom of choiceConvergenceCountingCyclic group

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