Cayley graph — the series

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.

part 1 · algebra

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.

part 2 · algebra

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.

part 3 · algebra

The free group on two generators with its middle removed: 4 pieces. The Cayley graph of the free group on two generators, with the elements within 0 steps of the identity greyed out and the remaining elements coloured by which connected piece they fall in.

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.

part 4 · algebra

One lattice, 3 generating sets, 3 shapes of ball. Lattice points reached within a fixed number of steps in the integers squared, for one step along either axis, axis steps and one diagonal, a king's moves, each drawn inside the polygon spanned by its steps and scaled by the radius.

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.

part 5 · algebra

A ball of the square grid with random travel times. A ragged roughly round region of grid cells shaded in bands by the time a signal from the centre first reaches them.

The shape a random ball grows into

Give every road of the square grid a random travel time and ask what can be reached from one point in time t. The region is ragged, and rescaled it converges to a fixed convex shape — but which shape is unknown for every natural law. Computing it shows a curve within a few per cent of a circle for continuous travel times, a flat side where fast roads percolate along a diagonal, a time per step that is still drifting at a hundred and twenty-eight steps, and fluctuations that grow like the distance to the power one third rather than one half.

part 6 · algebra

How often a walk comes home, on three lattices and a tree. Four curves of the logarithm of the return probability after 2n steps, for n up to 60: three lattices flattening, and the free group falling along a straight line.

How rarely a walk on a group comes home

Walk at random on the picture of a group, one generator at a time, and ask for the chance of standing at the start after 2n steps. On the line, the plane and three-dimensional space it falls like a power of n. On the tree that pictures the free group it falls by the factor √3/2 every step, exponentially. Kesten proved in 1959 that this is no accident of two examples: the chance falls exponentially exactly when the group's balls are mostly boundary, so a probabilistic rate and a geometric ratio are the same measurement.

part 7 · algebra