Cayley graph
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
Also named here as word metric — the same set of essays touches all of them, so they are one junction rather than several.
The group drawn as a map
A multiplication table says everything about a group and shows nothing; lay the same information out as one dot per element and one arrow per generator, and multiplying becomes walking, distance becomes a word length, and the group acquires a shape.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Group actionWord metricCyclic groupFree groupGenerating setGrowth rateInvariantAxiom of choiceCosetDihedral groupNon-measurable setSymmetry