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Concept

Ends of a group

The number of separate ways a group's picture runs off to infinity: remove a large finite piece of its Cayley graph and count the infinite pieces left. It is always 0, 1, 2 or infinitely many, and it records whether the group is finite, plane-like, a line up to finite fuzz, or splits along a finite subgroup.

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

Also named here as quasi isometry — the same set of essays touches all of them, so they are one junction rather than several.

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.

algebra · Cayley graph 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.

algebra · Cayley graph

Named alongside it

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

Cayley graphGrowth rateQuasi isometryWord metricConnectednessConvex hullFree groupFree productInvariantLatticeLimit shapeNorm

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