Algebraic Map Theory Gareth Jones School of Mathematics University - - PowerPoint PPT Presentation

Algebraic Map Theory

Gareth Jones

School of Mathematics University of Southampton UK

June 1, 2014

Outline of the talk

A group-theoretic approach to embeddings of graphs in surfaces (compact and oriented, for simplicity).

Outline of the talk

A group-theoretic approach to embeddings of graphs in surfaces (compact and oriented, for simplicity). Main topics:

I Regular maps: the most symmetric embeddings of various

classes of arc-transitive graphs (complete, complete bipartite, etc), where techniques from finite group theory yield classifications.

Outline of the talk

A group-theoretic approach to embeddings of graphs in surfaces (compact and oriented, for simplicity). Main topics:

I Regular maps: the most symmetric embeddings of various

classes of arc-transitive graphs (complete, complete bipartite, etc), where techniques from finite group theory yield classifications.

I Bipartite graph embeddings (called dessins d’enfants) provide

a bridge between the theories of Riemann surfaces and of algebraic number fields.

Maps

A map M is an embedding of a graph G (finite, connected, possibly with loops and multiple edges) in a surface S (compact, connected, oriented, without boundary), so that the faces (connected components of S \ G) are homeomorphic to discs. Example (the one-armed bandit) This is a map on the sphere; if you become bored during my talk, try to guess which mathematician it represents. A bipartite map B is a map in which G is bipartite.

Bipartite maps and permutations

A bipartite map B may be represented as a permutation group G = hx, yi (finite, transitive) on the set E of edges of B. e ex ey ez

• rientation

Here x and y rotate edges around their incident white and black vertices, following the orientation of the underlying surface S, while z := (xy)1 rotates edges two steps around faces. Warning: x, y and z are not generally automorphisms of B.

Bipartite maps

A bipartite map B may be represented as a permutation group G = hx, yi (finite, transitive) on the set E of edges of B. The black and white vertices and the faces correspond to the cycles of x, y and z on E. Conversely, any finite transitive 2-generator permutation group G = hx, yi determines a bipartite map B: incidence = non-empty intersection of cycles, cyclic order gives orientation. Example The natural representation of A5, with |x| = 3, |y| = 2 (so |z| = 5), gives the spherical map If B0 corresponds to G 0 = hx0, y0i then B ⇠ = B0 if and only if there is an isomorphism G ! G 0 with x 7! x0, y 7! y0.

Monodromy and automorphism groups

G (= hx, yi) is called the monodromy group of B. The automorphisms of B (preserving orientation and colour) are the permutations of E commuting with x and y (equivalently G). The automorphism group A = Aut B is the centraliser C(G) of G in the symmetric group Sym(E) on E. A acts semi-regularly on E (i.e. Ae = 1 for all e 2 E), and A ⇠ = NG(Ge)/Ge, where NG( ) denotes normaliser in G. B is regular if A is transitive on E. The following are equivalent:

I B is regular; I A acts regularly on E; I G acts regularly on E.

In this case A ⇠ = G (left and right regular representations of the same group), though A 6= G unless they are abelian.

Two regular examples

Figure : Regular sphere and torus embeddings of the cube graph Q3

On the right, identify opposite sides of the outer hexagon to form a torus. In each case A ⇠ = G ⇠ = A4 acting regularly, with x and y of order 3. On the left z has order 2, on the right it has order 3.

Characteristic, genus and type

The Euler characteristic of B is χ = |V | |E| + |F| = σ(x) + σ(y) + σ(z) σ(1), where σ( ) denotes number of cycles. The genus is g = 1 χ 2 . If x, y and z have orders l, m and n (= lcms of cycle-lengths) B has type (l, m, n). If B is regular then x, y and z have all their cycles of lengths l, m and n, so σ(x) = |E|/l, σ(y) = |E|/m and σ(z) = |E|/n, giving χ = |E| ✓1 l + 1 m + 1 n 1 ◆ .

Two regular examples, revisited

Type (3, 3, 2), χ = 2, g = 0. Type (3, 3, 3), χ = 0, g = 1.

Equal rights for non-bipartite maps

Any map M may be converted into a bipartite map B on the same surface, and described by a monodromy group G = hx, yi: divide each edge into two edges separated by a white vertex of valency 2, so x rotates arcs around vertices, y reverses arcs (Hamilton, 1856). B 7! M arcs of M ! edges of B.

Equal rights for non-bipartite maps

Any map M may be converted into a bipartite map B on the same surface, and described by a monodromy group G = hx, yi: divide each edge into two edges separated by a white vertex of valency 2, so x rotates arcs around vertices, y reverses arcs (Hamilton, 1856). B 7! M arcs of M ! edges of B. Example M = Monsieur Mathieu. G = M12 (Mathieu group), a sporadic simple group

• f order 95040,

⇠ = Aut S(5, 6, 12). ´ Emile L´ eonard Mathieu, 1835–1890.

The free group of rank 2

Any bipartite map B gives a finite transitive permutation representation ∆ ! G, X 7! x, Y 7! y, Z 7! z

• f the free group of rank 2

∆ = F2 = hX, Y , Z | XYZ = 1i = hX, Y | i. B corresponds to a conjugacy class of map subgroups M = ∆e (stabilisers of edges e 2 E) of finite index in ∆. B is regular if and only if M is normal in ∆, in which case A ⇠ = G ⇠ = ∆/M.

Examples

1 Taking M = ∆, so A = G = 1, gives the trivial bipartite map B1

• n the sphere, with graph K1,1 (one black vertex, one white vertex,
• ne edge and one face):
• 2 Hall (QJM 1935) showed that ∆ has 19 normal subgroups M

with ∆/M ⇠ = A5, so there are 19 regular bipartite maps B with automorphism group A ⇠ = A5. For instance the dodecahedron and icosahedron give B of type (3, 2, 5) and (5, 2, 3) and genus 0; the great dodecahedron gives B of type (5, 2, 5) and genus 4. 3 A5 has seven faithful transitive permutation representations, so there are 19 ⇥ 7 = 133 bipartite maps B with monodromy group G ⇠ = A5; they include the 19 regular maps above, and this non-regular spherical map shown earlier:

Coverings

Coverings (possibly branched) of maps B1 ! B2 correspond to inclusions M1  M2 of map subgroups in ∆. Normal inclusions induce regular coverings, by the subgroup M2/M1  N∆(M1)/M1 = Aut B1.

Theorem (Singerman and J, PLMS 1978)

Every bipartite map B is the quotient ˜ B/H of a regular bipartite map ˜ B of the same type by a subgroup H of Aut ˜ B.

• Proof. If B corresponds to M  ∆, let ˜

B be the regular map corresponding to the core ˜ M of M in ∆ (the largest normal subgroup of ∆ contained in M). Then ˜ B is regular since ˜ M is normal in ∆, it has the same type as B since they have the same monodromy group G = ∆/ ˜ M, and B ⇠ = ˜ B/H with H = M/ ˜ M  ∆/M ⇠ = Aut ˜ B.

Coverings

Theorem (Singerman and J, PLMS 1978)

Every bipartite map B is the quotient ˜ B/H of a regular bipartite map ˜ B of the same type by a subgroup H of Aut ˜ B. Hence it is (often) sufficient to study regular bipartite maps and their automorphism groups. Example This bipartite map B of type (3, 2, 5) is the quotient of ˜ B = dodecahedral map (made bipartite) by a subgroup H ⇠ = A4 of Aut ˜ B ⇠ = A5.

Another example

M If M is Monsieur Mathieu then ˜ M is a regular map of genus 3601 with Aut ˜ M ⇠ = M12, and M ⇠ = ˜ M/H where H ⇠ = M11 (the smallest Mathieu group).

Classifications

Using this machinery, one can try to classify regular maps by:

I Automorphism group A: classify generating pairs x, y for

G ⇠ = A, modulo the action of Aut G (e.g. A = A5 earlier).

I Surface S: Marston Conder’s website has computer-generated

lists up to genus 301, found by determining low-index subgroups of triangle groups.

I Graph G: find suitable arc-transitive subgroups A  Aut G

and suitable generating pairs x, y for G ⇠ = A; achieved for many classes, e.g. complete, complete multipartite, Hamming, Johnson, Paley, n-cubes, etc.

Complete graphs

Theorem (Biggs 1971; James and J, 1985)

Kn (n 2) has a regular embedding iff n = pe (p prime); there are φ(n 1)/e of them up to isomorphism, all with A ⇠ = AGL1(Fn) = {t 7! at + b | a, b 2 Fn, a 6= 0}. They are constructed (by Biggs) as Cayley maps over the group (Fn, +): take V = Fn, and let each v 2 V have neighbours v + 1, v + α, v + α2, . . . , v + αn2 in cyclic order, where F⇥

n = hαi.

Since F⇥

n ⇠

= Cn1 there are φ(n 1) choices for α; orbits of Gal Fn ⇠ = Ce on them correspond to isomorphism classes of maps. The proof uses Zassenhaus’s classification of sharply 2-transitive permutation groups (here A acting on V ).

Two regular torus embeddings of K5

In each case, identify opposite sides of the outer square to form a torus.

Two regular torus embeddings of K5

α = 2 α = 3

Complete bipartite graphs

Unlike Kn, Kn,n has at least one regular embedding for each n: the standard embedding is a Cayley map for (Z2n, +) with generating set 1, 3, . . . , 2n 1, so each v 2 V = Z2n has neighbours v + 1, v + 3, . . . , v + 2n 1 in that cyclic order, giving a regular map of genus (n 1)(n 2)/2. However, for most n there are also non-standard embeddings. Du, Kwak, Nedela, ˇ Skoviera, Zlatoˇ s and J classified the regular embeddings of Kn,n (2001–2010). The general result is too complicated to state here, but if n = pe (prime p > 2, e 1) there are pe1 of them, all of genus (n 1)(n 2)/2. The proof uses results of Huppert and Wielandt on groups which factorise as products of cyclic groups, and Hall’s generalisations of Sylow’s Theorems for finite solvable groups.

The standard embeddings of K3,3 and K4,4

(a) A A B B (b) In (a), identify opposite sides of the outer hexagon to form a torus. In (b), identify sides of the outer 16-gon, as indicated and using rotational symmetry by C4, to form a surface of genus 3.

A nonstandard regular embedding of K4,4

Identify opposite sides of the outer square to form a torus.

The universal bipartite map B1

This non-compact map corresponds to the subgroup M = 1 of ∆. Surface = hyperbolic plane H = {z 2 C | Im z > 0}. Vertices = rationals a/b, b odd; black or white as a is even or odd. Edge a/b to c/d (hyperbolic geodesic) iff ad bc = ±1. Face-centres a/b with b even (including 1 = 1/0).

1 1 1 2 1 1 3 2 3 4 3 5 3 1 5 2 5 3 5 4 5 6 5 7 5 8 5 9 5

Q H

Figure : Part of B∞ (for 0  Re z  2 and b  5); repeat with period 2.

Automorphisms of B1

1 1 1 2 1 1 3 2 3 4 3 5 3 1 5 2 5 3 5 4 5 6 5 7 5 8 5 9 5

X Y Aut B1 is a free group of rank 2, generated by M¨

• bius

transformations X : z 7! z 2z + 1 and Y : z 7! z 2 2z 3 fixing the black and white vertices at 0 and 1 and cyclicly rotating their incident edges. (This is the principal congruence subgroup Γ(2) of level 2 in the modular group Γ = PSL2(Z).)

Bely˘ ı’s Theorem

Any bipartite map B is isomorphic to B1/M for some M  ∆. B1 is on a Riemann surface H ⇢ C, so B1/M is on the Riemann surface H/M. Filling in punctures at the vertices and edge centres gives a compact Riemann surface X = (H [ Q [ {1})/M supporting B. Which Riemann surfaces arise in this way from maps? Not all, since for each genus g 1 there are uncountably many Riemann surfaces, but only countable many maps. Riemann: compact Riemann surface = complex algebraic curve. Bely˘ ı’s Theorem (1979), reformulated by Grothendieck & Wolfart: A compact Riemann surface is obtained from a bipartite map iff it is defined, as an algebraic curve, over Q = {algebraic numbers}.

Bely˘ ı curves and functions

Such a curve X, defined over Q, is called a Bely˘ ı curve. The inclusion M  ∆ induces a covering (of maps and surfaces) β : B = B1/M ! B1/∆ = B1 = • X ! P1(C) = S2, called a Bely˘ ı function. This is meromorphic (= rational), defined

• ver Q, unbranched outside {0, 1, 1}. Then B = β1(B1), with

{black vertices} = β1(0), {white vertices} = β1(1) {face centres} = β1(1), {edges} = β1([0, 1]) Grothendieck called such maps B dessins d’enfants (children’s drawings).

An example

Let B be the standard embedding of Kn,n, a regular bipartite map. Then M = ∆0∆n, generated by the commutators and nth powers, A = G ⇠ = ∆/M ⇠ = Cn ⇥ Cn. X is the nth degree Fermat curve (defined over Q ⇢ Q) Fn = {[x0, x1, x2] 2 P2(C) | xn

0 + xn 1 = xn 2 },

with Bely˘ ı function β : Fn ! P1(C), [x0, x1, x2] 7! ✓x0 x2 ◆n . Automorphisms of B multiply x0 and x1 by nth roots of 1.

Galois operations

A dessin may be identified with a Bely˘ ı pair (X, β), both defined

• ver the field Q of algebraic numbers.

The absolute Galois group G = Gal Q/Q = Aut Q acts on the coefficients of the equations defining X and β, inducing actions on Bely˘ ı pairs and hence on dessins. The group G is very important in algebraic number theory, but it is also very complicated and difficult to work with. In 1984 Grothendieck suggested studying G through its action on dessins (and other related structures).

An easy example

M M Monsieur and Madame Mathieu M and M are defined over K = Q(p11). They are transposed by the Galois group GK = Gal K/Q ⇠ = C2 (complex conjugation) of K, forming an orbit of G of length 2.

A less obvious example

These three dessins are defined over the splitting field K of the polynomial f (t) = 25t3 12t2 24t 16, and are permuted transitively by its Galois group GK = Gal K/Q ⇠ = S3, forming an

• rbit of G of length 3.

Invariants of G

The following properties of a dessin can be defined algebraically, and are therefore invariant under G (Streit and J, 1997):

I number of edges; I valency distributions of white and black vertices and faces; I type and genus; I monodromy group and automorphism group.

Nevertheless, G acts faithfully on (isomorphism classes of)

I dessins (Grothendieck); I dessins of a given genus (Girondo and Gonz´

alez-Diez);

I plane trees = maps of genus 0 with one face (Schneps); I regular dessins (Gonz´

alez-Diez and Jaikin-Zapirain);

I regular dessins of a given hyperbolic type (Kucharczyk).

Faithful action of G

G acts faithfully on (isomorphism classes of)

I dessins (Grothendieck); I dessins of a given genus (Girondo and Gonz´

alez-Diez);

I plane trees = maps of genus 0 with one face (Schneps); I regular dessins (Gonz´

alez-Diez and Jaikin-Zapirain);

I regular dessins of a given hyperbolic type (Kucharczyk).

Consequence: in principle, one can ‘see’ all of algebraic number theory by looking at any one of the above classes of dessins. Practical problem: it is very difficult to give explicit examples of

• rbits of G on dessins which reveal much of its structure.

Regular dessins based on Kn and Kn,n

Using map operations introduced by Wilson (1979), Streit, Wolfart and J (PLMS 2010) found the following orbits of G on regular dessins: The φ(n 1)/e regular dessins which embed Kn (n = pe) are all defined over a particular subfield K of the (n 1)th cyclotomic field Q(

n−1

p 1), and are permuted transitively by GK. The n/p = pe1 regular dessins which embed Kn,n (n = pe odd) are all defined over the (n/p)th cyclotomic field K = Q(

n/p

p 1), and form orbits of lengths φ(pi) (i = 0, 1, . . . , e 1) under GK. On each of these orbits G induces an abelian group, so the commutator subgroup G0 is in the kernel of the action. Problem: find large explicit orbits exhibiting non-abelian actions.

Structure of G

Q = [

K2K

K, where K is the set of Galois (finite, normal) extensions of Q in C. For each K 2 K let GK := Gal K/Q = Aut K, a finite group. If K L in K there is a restriction epimorphism ρK,L : GK ! GL. Then G = lim

GK,

a profinite group (= projective limit of finite groups). Specifically, G = {(gK) 2 Y

K2K

GK | ρK,L(gK) = gL whenever K L}.

Topology on G

G = {(gK) 2 Q

K2K GK | ρK,L(gK) = gL if K L} is uncountable.

. If we put the discrete topology on each GK then Q

K2K GK is a

topological group, compact by Tychonoff’s Theorem. As a closed subgroup, G is also compact in the induced Krull

• topology. (Two elements are ‘close’ if they agree on a large

subfield of Q.) The topology is that of a Cantor set. In the Galois correspondence, subfields of Q correspond to closed subgroups of G. Hilbert’s conjecture that every finite group F is a Galois group over Q is equivalent to showing that F is a quotient of G by a closed normal subgroup. This has been proved for many F (e.g. solvable, symmetric or alternating), but it is still open in general.