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 = h x , y i (finite, transitive) on the set E of edges of B . orientation ez ex e ey 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 = h x , y i (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 = h x , y i determines a bipartite map B : incidence = non-empty intersection of cycles, cyclic order gives orientation. Example The natural representation of A 5 , with | x | = 3 , | y | = 2 (so | z | = 5), gives the spherical map If B 0 corresponds to G 0 = h x 0 , y 0 i then B ⇠ = B 0 if and only if there is an isomorphism G ! G 0 with x 7! x 0 , y 7! y 0 .

Monodromy and automorphism groups G (= h x , y i ) 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. A e = 1 for all e 2 E ), and A ⇠ = N G ( G e ) / G e , where N G ( ) 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 Q 3 On the right, identify opposite sides of the outer hexagon to form a torus. In each case A ⇠ = G ⇠ = A 4 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 ✓ 1 l + 1 m + 1 ◆ χ = | E | 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 = h x , y i : 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). 7! M B 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 = h x , y i : 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). 7! M B arcs of M ! edges of B . Example M = Monsieur Mathieu. G = M 12 (Mathieu group), a sporadic simple group of 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 of the free group of rank 2 ∆ = F 2 = h X , Y , Z | XYZ = 1 i = h X , 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 B 1 on the sphere, with graph K 1 , 1 (one black vertex, one white vertex, one edge and one face): • �� 2 Hall (QJM 1935) showed that ∆ has 19 normal subgroups M with ∆ / M ⇠ = A 5 , so there are 19 regular bipartite maps B with automorphism group A ⇠ = A 5 . 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 A 5 has seven faithful transitive permutation representations, so there are 19 ⇥ 7 = 133 bipartite maps B with monodromy group G ⇠ = A 5 ; they include the 19 regular maps above, and this non-regular spherical map shown earlier:

Coverings Coverings (possibly branched) of maps B 1 ! B 2 correspond to inclusions M 1  M 2 of map subgroups in ∆ . Normal inclusions induce regular coverings, by the subgroup M 2 / M 1  N ∆ ( M 1 ) / M 1 = Aut B 1 . 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 ⇠ = ˜ H = M / ˜ M  ∆ / M ⇠ = Aut ˜ B / H with 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) su ffi cient 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 ⇠ = A 4 of B ⇠ Aut ˜ = A 5 .

Another example M ˜ If M is Monsieur Mathieu then M is a regular map of genus 3601 M ⇠ = M 12 , and M ⇠ M / H where H ⇠ with Aut ˜ = ˜ = M 11 (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 = A 5 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) K n ( n � 2) has a regular embedding i ff n = p e (p prime); there are φ ( n � 1) / e of them up to isomorphism, all with A ⇠ = AGL 1 ( F n ) = { t 7! at + b | a , b 2 F n , a 6 = 0 } . They are constructed (by Biggs) as Cayley maps over the group ( F n , +): take V = F n , and let each v 2 V have neighbours v + 1 , v + α , v + α 2 , . . . , v + α n � 2 in cyclic order, where F ⇥ n = h α i . n ⇠ Since F ⇥ = C n � 1 there are φ ( n � 1) choices for α ; orbits of Gal F n ⇠ = C e 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 K 5 In each case, identify opposite sides of the outer square to form a torus.

Two regular torus embeddings of K 5 α = 2 α = 3