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Concept

Free group

The group of all reduced words in a set of letters and their inverses, with no relations beyond a letter cancelling against its own inverse. It is the fundamental group of a wedge of circles, and every group generated by that many elements is a quotient of it.

Named by 12 essays across 2 fields — each of them below, with the objects they name alongside it.

A wedge of 2 circles. Several circles all passing through one common point, each labelled with a generator, so that a loop is a word in those letters.

The subgroup that is freer than the group

A free group on two letters contains a subgroup of index three that is free on four. Nothing about a group makes that plausible; everything about a graph makes it obvious, and the argument is to stop looking at the group and start looking at the space whose loops it is.

topology · Covering spaces the Borromean rings, with every crossing signed. A diagram of the Borromean rings with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

Linked, and no two of them are

Three rings that cannot be pulled apart, in which every pair comes apart the moment the third is removed. Every pairwise linking number is zero, so the number cannot see it — and what does see it is a word in two letters that refuses to cancel.

topology · Linking number Three spaces, cut into pieces. 3 panels, drawn from: a sphere split into two caps meeting along a circle, two circles joined at a point, and a square whose opposite edges are glued into a torus. Each carries the fundamental group that van Kampen's theorem computes for it.

Cutting a space to find its group

A space assembled from two pieces has a fundamental group assembled from theirs, and the recipe is exact — take everything both groups offer and impose the relations the overlap forces. Almost every fundamental group anybody knows is computed this way, including all of the surfaces.

topology · Homotopy 3 symmetries over 3 sheets: a regular covering. A 3-sheeted covering of a wedge of 2 circles, with the permutations of its sheets that commute with every generator. There are 3, against 3 sheets.

The symmetries a cover has of its own

A covering space can be shuffled without disturbing anything below it, and how many ways there are is decided by the subgroup it corresponds to. When there are as many symmetries as sheets the covering is called regular, and that is the same statement as the subgroup being normal.

topology · Covering spaces 3 sheets, 8 of 26 words coming back. A table of reduced words in two generators with the sheet each sends the base sheet to. The words returning to it are the covering's subgroup, and the 3 sheets are its cosets.

A covering is a permutation

Describing a covering means saying where each loop sends each sheet, which is a permutation for every generator. So a covering of a wedge of circles is nothing but a homomorphism to a symmetric group, and the subgroup it corresponds to is a stabiliser.

topology · Covering spaces 6 vertices folded to 4, and a graph that decides. The graph built from 3 generator words, folded until no vertex has two edges of one label leaving it. Reading a word from the base vertex decides membership, and 6 words are tested.

Folding a graph until it decides

A subgroup of a free group usually arrives as a list of words, and almost nothing about it is readable from the list. Draw the words as loops, merge every pair of edges with the same label leaving one point, and what is left is a machine that decides membership by reading.

topology · Covering spaces 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 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 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.

algebra · Cayley graph A loop that cannot be undone, and the surface it bounds. The loop aba⁻¹b⁻¹ on the figure eight, which cannot be pulled tight, beside a square whose boundary reads the same word, showing that the loop bounds a surface.

What homology forgets about a loop

Let the letters of a loop commute and the loop group of a space becomes its first homology group: a loop now records only how often it went round each hole. What is thrown away is exactly the loops that bound a surface. On the figure eight that is nearly everything — of the loops of sixty letters that homology calls nought, about one in 6,700 is a loop that actually shrinks.

topology · Homotopy A random walk winding round a puncture. Lattice walk of 4000 steps around (½, ½); final winding -4.861 turns; box -12..26 × -29..22.

How a random path winds round a point

A path in the plane with one point removed has a class in the punctured plane's fundamental group: how many times it has wound round the point. For a random path the class grows like the logarithm of time, and its spread has a law. Brownian motion follows Spitzer's Cauchy law, so heavy-tailed that the mean winding does not exist; a walk on a grid follows the hyperbolic secant law instead, with tails that fall exponentially. The difference is made entirely by close passes, which a grid forbids. With two points removed the class becomes a word, and some loops wind round neither point and still cannot be shrunk.

topology · Homotopy

Named alongside it

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

Fundamental groupCovering spaceCayley graphGrowth rateSubgroupGraphGroup actionIndexInvariantRandom walkWord metricCommutator

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