Automorphism groups of edge-transitive maps (Pilsen 2016) Gareth - - PowerPoint PPT Presentation

Automorphism groups of edge-transitive maps (Pilsen 2016)

Gareth Jones

University of Southampton, UK

October 9, 2016

Recent history

In 1997 Graver and Watkins showed how edge-transitive maps M can be partitioned into 14 classes T. These are distinguished by the isomorphism class N(T) of the one-edge map M/Aut M. In 2001 ˇ Sir´ aˇ n, Tucker and Watkins showed that for each n ≥ 11 with n ≡ 3 or 11 mod (12), there are finite, orientable, edge-transitive maps M in each class T with Aut M ∼ = Sn. In 2011 Orbaniˇ c, Pellicer, Pisanski and Tucker classified the edge-transitive maps of low genus, together with those on E2. Karab´ aˇ s and Nedela (work in progress) have introduced a similar partition of oriented edge-transitive maps, based on M/Aut+M, which allows them to extend the classifications to higher genus. I shall consider which groups can arise as Aut M for maps M, finite or infinite, with or without boundary, in the various classes T.

• G. A. Jones, Automorphism groups of edge-transitive maps,

arXiv.math [CO] 1605.09461.

Maps

A map M is an embedding of a graph G in a surface S, such that the faces (connected components of S \ G) are simply connected, i.e. homeomorphic to an open disc. The regular (or Platonic) solids are typical examples. I shall assume that S and G are connected; S may be orientable or not, compact or not, with or without boundary (generally without). The graph G may have multiple edges, loops and half-edges (though not usually in the most symmetric cases which I will concentrate on). An automorphism of M is an automorphism of G which extends to a self-homeomorphism of S. These form a group Aut M. Problem Which groups arise as the automorphism groups of highly symmetric maps?

Maps and permutations

v e f φ φr0 φr1 φr2 φr0r2 The monodromy group G = r0, r1, r2 | r2

i = (r0r2)2 = 1, . . .

• f a map M acts transitively on the set Φ of flags φ = (v, e, f ) of

M, with ri changing the i-dimensional component of each φ while preserving the other two. Vertices, edges and faces correspond to

• rbits of r1, r2, r0, r2 (∼

= V4) and r0, r1 on Φ. The automorphism group A = Aut M of M is the centraliser of G in the symmetric group Sym Φ, acting semiregularly on Φ.

Map subgroups

Maps M correspond to transitive permutation representations of Γ := R0, R1, R2 | R2

i = (R0R2)2 = 1,

via epimorphisms Γ → G, Ri → ri (i = 0, 1, 2), and hence to conjugacy classes of map subgroups M = Γφ = {γ ∈ Γ | φγ = φ} ≤ Γ (φ ∈ Φ). Easy arguments show that

• 1. Aut M ∼

= NΓ(M)/M,

• 2. Aut M acts transitively on Φ if and only if M is normal in Γ,

in which case Aut M ∼ = Γ/M ∼ = G, all acting regularly on Φ. Such maps M are called regular.

Regular maps and their groups

Regular maps are the most symmetric, the most studied, and the most important of all maps. For example, every map is the quotient of a regular map by some group of automorphisms. For a given group G, the regular maps M with Aut M ∼ = G correspond to the normal subgroups M of Γ with Γ/M ∼ = G. If G is finite, the number of them is |Epi (Γ, G)|/|Aut G|. Problem Which groups G are automorphism groups of regular maps? Equivalently, which groups G are quotients of Γ = R0, R1, R2 | R2

i = (R0R2)2 = 1 ?

Note that Γ = R0, R2 ∗ R1 ∼ = V4 ∗ C2, the free product of a Klein four-group and a cyclic group of order 2.

Finite simple groups

As a starting point, one could consider the finite simple groups as candidates for automorphism groups of regular maps. The classification of finite simple groups (CFSG), announced around 1981, is as follows:

◮ cyclic groups Cp, p prime, ◮ alternating groups An, n ≥ 5, ◮ finitely many infinite families of groups of Lie type, defined

• ver finite fields Fq, such as

◮ projective special linear groups Ln(q) = PSLn(q), n ≥ 2, ◮ unitary groups Un(q), n ≥ 3, etc,

◮ 26 sporadic groups, such as the Mathieu groups Mn (acting

• n Steiner systems) for n = 11, 12, 22, 23, 24, the McLaughlin

group McL (acting on SRG(275, 112, 30, 56)), etc.

Regular maps and Mazurov’s question

In 1980 Mazurov asked in the Kourovka Notebook (Problem 7.30): which finite simple groups are generated by three involutions, two

• f them commuting, i.e. which of them are quotients of Γ?

It is now known from work of Nuzhin and others that all non-abelian finite simple groups have such generators, except:

◮ L3(q) (:= PSL3(q)) and U3(q) for all prime powers q, ◮ L4(q) and U4(q) for q = 2e, ◮ A6, A7, M11, M22, M23 and McL.

Note that these exceptions include L2(7) ∼ = L3(2), L2(9) ∼ = A6 and A8 ∼ = L4(2). (See surveys by Mazurov or ˇ Sir´ aˇ n for references.) Thus, apart from these exceptions, every non-abelian finite simple group is the automorphism group of a regular map. Indeed, for some groups one can count, and even classify, the associated maps. [A recent computer search by Martin Maˇ caj, independently confirmed by Matan Ziv-Av, suggests that U4(3) and U5(2) should be added to the list of exceptions.]

Example 1: G = A5, of order 60

Look for epimorphisms Γ = V4 ∗ C2 → G. The factors V4 and C2 must be embedded in G. There are 15 involutions in G, each commuting with two others, so there are 30 embeddings V4 → G. There are three involutions in any subgroup V ∼ = V4, leaving 12 involutions outside it. The only maximal subgroup containing V is its normaliser, a subgroup A ∼ = A4, which contains no further involutions. Hence any of the remaining 12 involutions, together with V , generates G, so there are 30.12 = 360 epimorphisms Γ → G. AutG = S5 permutes these epimorphisms regularly, so there are 360/5! = 3 normal subgroups N ⊳ Γ with Γ/N ∼ = G. Thus there are three regular maps M with Aut M ∼ = A5. They are the antipodal quotients of the icosahedron, dodecahedron and great dodecahedron, non-orientable maps of genus 1, 1 and 5.

Example 2: G = L3(2) (∼ = L2(7)), of order 168

This is the second smallest non-abelian finite simple group. It is the automorphism group of the Fano plane. Similar counting arguments show that the images of V4 and C2 in G always lie in a proper subgroup (leaving a point or a line invariant), so they cannot generate G. Hence there are no regular maps M with Aut M ∼ = L3(2).

Edge-transitive maps

What about a wider set of highly symmetric maps, namely edge-transitive maps? The following is easy to prove:

Lemma

Aut M acts transitively on the edges of M if and only if Γ = NE, where N := NΓ(M) and E := R0, R2 ∼ = V4. Since |E| = 4 this implies that |Γ : N| ≤ 4. By inspection there are just 14 conjugacy classes of subgroups N ≤ Γ satisfying Γ = NE. They correspond to the 14 possible maps M/Aut M with one edge, and to the 14 classes of edge-transitive maps M described by Graver and Watkins in 1997 (Mem. Amer. Math. Soc. 601). Example Class 1 consists of the regular maps, those with N = Γ. These include the Platonic solids, the antipodal quotients of the cube, octahedron, dodecahedron and icosahedron, and many more.

1 2 2∗ 2P 2ex 2∗ex 2Pex 3 4 4∗ 4P 5 5∗ 5P = closed disc = sphere = M¨

• bius band

= real projective plane Basic maps N(T) = M/Aut M for the edge-transitive classes T.

Example: the cube

The cube, as a map M on the sphere, has Aut M ∼ = S4 × C2. It is regular, hence vertex-, edge-, and face-transitive.

F M F N(1) The cube M satisfies M/Aut M ∼ = F ∼ = N(1), where F is a fundamental region for Aut M, so M is in class 1.

Example: the cuboctahedron

The cuboctahedron, as a map M on the sphere, also has Aut M ∼ = S4 × C2. It is edge- and vertex-transitive, but not face-transitive.

M F F N(2∗) The cuboctahedron M satisfies M/Aut M ∼ = F ∼ = N(2∗), where F is a fundamental region for Aut M, so M is in class 2∗.

Orientably regular chiral maps

Class 2Pex (blame Jack Graver and Mark Watkins for the notation!) consists of those maps for which N is the even subgroup Γ+ = X = R1R2, Y = R2R0 | Y 2 = 1 ∼ = C∞ ∗ C2

• f index 2 in Γ, consisting of the words of even length in the Ri.

These maps M are orientable and without boundary. They are orientably regular (Aut M is transitive on directed edges) and chiral (have no orientation-reversing automorphisms), so they

• ccur in chiral (mirror-image) pairs M and M.

Example Opposite sides of the outer squares are identified to form a chiral pair of torus maps, with Aut M ∼ = AGL1(5) ∼ = C5 ⋊ C4.

Example, continued

M F N(2Pex) Since M/Aut M ∼ = N(2Pex), the map M is in class 2Pex.

Automorphism groups of orientably regular chiral maps

The automorphism groups G = Aut M of the maps M in class 2Pex are the quotients of Γ+ = X, Y | Y 2 = 1 by subgroups M which are normal in Γ+ but not in Γ. This is equivalent to

• 1. G = x, y | y2 = 1, . . ., and
• 2. no automorphism of G inverts x and fixes y.

It is known that every finite simple group has a generating pair satisfying (1), but what about (2)? The abelian groups Cp all fail (2), but in the non-abelian case we have:

Theorem (Leemans & Liebeck, 2016)

A non-abelian finite simple group G has a generating pair x, y satisfying (1) and (2) if and only if G is not isomorphic to L2(q), L3(q), U3(q) or A7 (q any prime power). Thus all except L2(q), L3(q), U3(q) and A7 are automorphism groups of orientably regular chiral maps.

Further questions

This deals with the finite simple groups realised by the classes T = 1 (regular maps) and 2Pex (orientably regular chiral maps), the classes most frequently studied. However:

◮ what about the other twelve Graver-Watkins classes of

edge-transitive maps? (Vertex- and face-transitivity are too weak conditions to expect reasonable results.)

◮ what about other groups, besides finite simple groups,

e.g. infinite groups (corresponding to non-compact maps)? For each class T of edge-transitive maps, define G(T) = {G | G ∼ = Aut M for some M ∈ T}, and G+(T) = {G | G ∼ = Aut M for some orientable map without boundary M ∈ T}

Operations on maps

The 14 classes T correspond to 14 conjugacy classes of subgroups N = N(T) ≤ Γ ∼ = V4 ∗ C2 (those with Γ = NE). The outer automorphism group Out Γ ∼ = Aut V4 ∼ = S3

• f Γ corresponds to Wilson’s group D, P ∼

= S3 of map operations, where D = duality and P = Petrie duality (preserving the graph, but transposing faces and Petrie polygons = closed zig-zag walks). Out Γ permutes these 14 conjugacy classes, in six orbits. Hence Out Γ permutes the sets G(T) of automorphism groups N(T)/M realised by these classes, so it is sufficient to consider one representative class T from each orbit.

← D → ← P → The six rows are the orbits

• f Out Γ = D, P ∼

= S3 on the 14 basic maps N(T). The operations D and P transpose entries in the second column with those in the first and third, but fix N(1) and N(3). 1 2 2∗ 2P 2ex 2∗ex 2Pex 3 4 4∗ 4P 5 5∗ 5P

Theorem

A non-abelian finite simple group is isomorphic to the automorphism group of an edge-transitive map in a class T if and

• nly if it is not one of the exceptions listed in the corresponding

row of the following table. Class T Non-abelian finite simple groups G ∈ G(T) 1 L3(q), U3(q), L4(2e), U4(2e), A6, A7, M11, M22, M23, McL (Nuzhin et al.) 2, 2∗, 2P U3(3) 2 ex, 2∗ex, 2Pex L2(q), L3(q), U3(q), A7 (Leemans & Liebeck) 3 none 4, 4∗, 4P none 5, 5∗, 5P L2(q)

Table : Non-abelian finite simple groups not in sets G(T).

Theorem

A symmetric group Sn, an alternating group An, or a projective special linear group L2(q) is isomorphic to the automorphism group of an edge-transitive map in a class T if and only if it satisfies the corresponding condition in the following table. Class T Sn An L2(q) 1 n ≥ 1 n = 1, 2, 5 or n ≥ 9 q = 3, 7, 9 2, 2∗, 2P n ≥ 2 n ≥ 5 q = 3 2 ex, 2∗ex, 2Pex n ≥ 6 n ≥ 8 no q 3 n ≥ 2 n ≥ 5 q = 3 4, 4∗, 4P n ≥ 2 n ≥ 4 every q 5, 5∗, 5P n ≥ 6 n ≥ 7 no q

Table : Groups Sn, An and L2(q) in sets G(T).

Theorem

A symmetric group Sn or an alternating group An is isomorphic to the automorphism group of an edge-transitive orientable map without boundary in a class T if and only if it satisfies the corresponding condition in the following table. Class T Sn ∈ G+(T) An ∈ G+(T) 1 n = 1, 5, 6 no n 2, 2∗ n = 1, 2, 5, 6 no n 2P n ≥ 3 no n 2 ex, 2∗ex n ≥ 7 no n 2Pex n ≥ 6 n ≥ 8 3 n ≥ 3 no n 4, 4∗, 4P n ≥ 3 no n 5, 5∗ n ≥ 6 n ≥ 7 5P n ≥ 6 no n

Table : Groups Sn and An in sets G+(T).

Theorem

There is a finite group of nilpotence class c, or of derived length l, isomorphic to the automorphism group of an edge-transitive map in a class T, if and only if c or l satisfy the corresponding condition in the following table. Class T Nilpotence class c Derived length l 2 ex, 2∗ex, 2Pex c ≥ 5 l ≥ 2 5, 5∗, 5P c ≥ 2 l ≥ 2 All other T c ≥ 1 l ≥ 1

Table : Nilpotence class and derived length.

Embedding theorems

Not every finite group G can be the automorphism group of an edge-transitive map: the normalisers N = N(T) ≤ Γ all have at most four generators, and hence so do their quotients G = N/M. However, given any class T, we have Sn ∈ G(T) for all sufficiently large n, so by Cayley’s Theorem each finite group G can be embedded in Aut M for some finite map M in T. If we allow infinite maps, then by using an embedding theorem of Schupp (1976) we have an analogue for countable groups:

Theorem

For each edge-transitive class T, every countable group G is isomorphic to a subgroup of Aut M for some map M in T.

Uncountably many maps and groups

By adapting B. H. Neumann’s proof (1937) that there are 2ℵ0 non-isomorphic 2-generator groups (realised as quotients of C∞ ∗ C3), we have:

Theorem

Each class T contains 2ℵ0 maps M with mutually non-isomorphic automorphism groups Aut M. (Of course, these maps M are also mutually non-isomorphic.) Grigorchuk’s groups with intermediate growth (1980–84) imply:

Theorem

One can choose these maps M so that M and Aut M have intermediate growth. In these maps, the numbers of vertices, edges and faces within distance d of a base-point grow faster than polynomially but slower than exponentially as d → ∞. (Most familiar examples of non-compact maps have polynomial or exponential growth.)

Decidability

A property P of groups is called a Markov property if

◮ it is preserved by isomorphisms, ◮ there is a finitely presented group with property P, and ◮ there is a finitely presented group which cannot be embedded

in any finitely presented group with property P. If P is inherited by subgroups the last two are equivalent to

◮ there are finitely presented groups with and without P.

Examples: being trivial, finite, nilpotent, solvable, free, torsion-free.

Theorem

For each class T of edge-transitive maps, and each Markov property P of groups, it is undecidable whether or not adding a finite set of relations to the standard presentation of N(T) produces a quotient group with property P.

• For example, it is undecidable whether adding a finite set of

relations to those of N(T) yields a trivial or a compact map.

Method of proof of the theorems

The 14 classes T correspond to 14 conjugacy classes of subgroups N(T) ≤ Γ ∼ = V4 ∗ C2. The outer automorphism group Out Γ ∼ = Aut V4 ∼ = S3

• f Γ, corresponding to Wilson’s group D, P of map operations,

has six orbits on these conjugacy classes, and it is usually sufficient to consider one representative of each orbit. Their presentations and structures are given by the Reidemeister-Schreier algorithm:

◮ N(1) = Γ ∼

= V4 ∗ C2 (regular maps);

◮ N(2Pex) = Γ+ ∼

= C∞ ∗ C2 (chiral maps);

◮ N(2) ∼

= C2 ∗ C2 ∗ C2;

◮ N(3) ∼

= C2 ∗ C2 ∗ C2 ∗ C2 (just-edge-transitive maps);

◮ N(4) ∼

= C∞ ∗ C2 ∗ C2;

◮ N(5) ∼

= C∞ ∗ C∞ ∼ = F2 (free group of rank 2).

Realising automorphism groups

To realise a given group G as Aut M for a map M in a class T, we need G ∼ = N(T)/M, where N(T) = NΓ(M), that is, M is normal in N(T) but not normal in any N(T ′) > N(T). This means that G must not have any ‘forbidden automorphisms’ arising from conjugation in N(T ′). These are as follows (with x, y, z, . . . generating successive cyclic factors of N(T)):

◮ N(1) = Γ ∼

= V4 ∗ C2: no forbidden automorphisms;

◮ N(2Pex) = Γ+ ∼

= C∞ ∗ C2: x → x−1, y → y;

◮ N(2) ∼

= C2 ∗ C2 ∗ C2: x ↔ y, z → z;

◮ N(3) ∼

= C2 ∗ C2 ∗ C2 ∗ C2: all three double transpositions;

◮ N(4) ∼

= C∞ ∗ C2 ∗ C2: x → x−1, y ↔ z;

◮ N(5) ∼

= C∞ ∗ C∞: transposing and/or inverting x and y. Finding quotients G of N(T) is fairly easy; avoiding forbidden automorphisms is generally much harder.

Reducing the problem

One can choose epimorphisms onto N(1) = Γ ∼ = V4 ∗ C2 from N(2) ∼ = C2 ∗C2 ∗C2, N(3) ∼ = C2 ∗C2 ∗C2 ∗C2, N(4) ∼ = C∞ ∗C2 ∗C2, which ensure (by composition) that if G is a quotient of Γ then it is a quotient, without forbidden automorphisms, of these three groups, so G is also realised for these three classes. Similarly, an epimorphism from N(5) ∼ = C∞ ∗ C∞ onto N(2Pex) = Γ+ ∼ = C∞ ∗ C2 ensures that any G realised for class 2Pex is also realised for class 5. This focuses attention on classes 1 and 2Pex (the regular and chiral maps, those of most interest combinatorially). However, some groups G can be realised for other classes 2, 3, 4 or 5 but not for 1 or 2Pex, so these four classes cannot be ignored.

Edge-transitive maps with boundary

A map M has non-empty boundary iff the map subgroup M contains a reflection (conjugate of some Ri). The regular maps with non-empty boundary are known (J, 2016). What about edge-transitive maps? Call a class T

◮ void if it contains no maps with non-empty boundary, ◮ tame if it contains such maps, all with dihedral groups, ◮ wild if it contains such maps, some with non-dihedral groups.

Theorem

◮ Classes 2 ex, 2∗ex, 2Pex, 5, 5∗ and 5P are void, ◮ classes 1, 2, 2∗ and 2P are tame, and ◮ classes 3, 4, 4∗ and 4P are wild.

For the tame classes 1, 2 and 2∗, all the maps arising are on the closed disc, while those in class 2P are on the closed disc, annulus, M¨

• bius band or infinite strip.

The regular maps with non-empty boundary

Two mutually dual infinite families etc Six small sporadic maps Two mutually dual infinite maps

Examples of wild behaviour

Results of Malle, Saxl and Weigel (1994) on generators of finite simple groups imply:

Theorem

If T is wild, then every non-abelian finite simple group is the automorphism group of a map with boundary in T, with the exception of U3(3) if T = 3. (U3(3), of order 6048, is the only non-abelian finite simple group not generated by three involutions.) Methods similar to those for maps without boundary give:

Theorem

If T is wild, then (a) given any countable group G, T contains a map M with boundary such that G is isomorphic to a subgroup of Aut M; (b) T contains 2ℵ0 maps with boundary, with mutually non-isomorphic automorphism groups.

Areas for future research

◮ Which non-simple finite groups are in the various sets G(T)?

(This is already known for Sn.) Given any finite group G ∈ G(T), a modified ‘Macbeath trick’ provides infinitely many finite covers ˜ G ∈ G(T), but one should expect many other covers to arise.

◮ Enumeration: given a finite group G ∈ G(T), how many maps

in T have automorphism group G? Can the well-known counting techniques for regular and

• rientably regular maps be adapted for use here?

◮ Asymptotics: how ‘popular’ are the various classes T among

finite edge-transitive maps? Do any classes predominate?

◮ Topology: which surfaces support maps in which classes T?

What can be said about orientability, genus, and number of boundary components?

Dˇ ekuji za poslech!