Word metric
Named by 5 essays across one field — each of them below, with the objects they name alongside it.
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.
What is left when the middle is taken out
Cut a finite piece out of a group's picture and count the parts of what remains that run off forever. The integers leave two, the plane one, a tree more with every cut — and no group anywhere leaves exactly three, because a third end is always the first of infinitely many.
The polygon a lattice becomes from far away
Walk the grid of whole-number points with a fixed set of moves and the places reachable in r moves fill a shape. With axis steps it is a diamond, add a diagonal and it is a hexagon, move like a knight and it is a ragged thing full of holes — which, seen from far enough away, is an octagon exactly. The generators decide the polygon, and the polygon decides the count.
Named alongside it
The objects these essays reach for when they reach for this one.
Cayley graphGrowth rateFree groupGroup actionInvariantCyclic groupEnds of a groupGenerating setQuasi isometryAxiom of choiceConnectednessConvex hull