Maximal subgroups of triangle groups Gareth Jones University of - - PowerPoint PPT Presentation

Maximal subgroups of triangle groups

Gareth Jones

University of Southampton, UK Summer School for Inetnational conference and PhD-Master on Groups and Graphs, Desighs and Dynamics

August 15, 2009

Outline of the talk

Triangle groups are important for finite and infinite group theorists, as well as those working in other areas such as Riemann surfaces or maps on surfaces. A great amount is known about their subgroups of finite index and their finite quotient groups, but less seems to be known about their subgroups of infinite index and their infinite quotient groups. I will describe some simple constructions of uncountably many conjugacy classes of maximal subgroups (mostly of infinite index) in various hyperbolic triangle groups, generalising results of Neumann, Magnus and others for the modular group. The constructions have applications to the realisation of groups as automorphism groups of maps and hypermaps on surfaces, giving analogues of results of Frucht and Sabidussi for graphs.

In 1933 B. H. Neumann used permutations to construct uncountably many subgroups of SL2(Z) which act regularly on the primitive elements of Z2, those (u, v) ∈ Z2 with u and v coprime. As pointed out by Magnus (1973, 1974), the images of these subgroups in the modular group Γ = PSL2(Z) = SL2(Z)/{±I} ∼ = C3 ∗ C2 are maximal nonparabolic subgroups, that is, maximal with respect to containing no parabolic elements. (Γ acts by M¨

• bius transformations on the upper half plane U ⊂ C

and on the rational projective line P1(Q) = Q ∪ {∞}. A non-identity element of Γ is parabolic if it has a fixed point in P1(Q), or equivalently has trace ±2; the parabolic elements of Γ are the conjugates of the powers Z i (i = 0) of Z : t → t + 1.) Further examples of maximal nonparabolic subgroups of Γ were subsequently found by Tretkoff and by Brenner and Lyndon.

The modular group and cubic maps

Let M be a map (a connected graph, embedded without crossings, and with simply connected faces) on an oriented surface. Assume that M is cubic (all vertex valencies divide 3), and allow free

• edges. Let Ω be the set of arcs (directed edges) of M.

Since Γ = X, Y , Z | X 3 = Y 2 = XYZ = 1, one can define a transitive action of Γ on Ω by letting X rotate arcs around their incident vertices (following the orientation), and Y reverse arcs, so 1-valent vertices and free edges give fixed points of X and Y . α αY αX αX 2 αZ = orientation Then vertices, edges and faces correspond to cycles of X, Y and Z.

For any map M (cubic, oriented), the map subgroups M = Γα = {g ∈ Γ | αg = α} (α ∈ Ω) are mutually conjugate. They have index |Γ : M| = |Ω|, and are maximal if and only Γ acts primitively on Ω, i.e. preserves no non-trivial equivalence relations on Ω.

Proposition

The map subgroups for the following map M are maximal in Γ and are non-parabolic. (Note: ‘maximal and nonparabolic’ = ‘maximal nonparabolic’!) M

• Proof. Here is M, with α the left-most arc. There is a unique face,

so Z has a single cycle on Ω, with each arc αZ i ∈ Ω labelled i ∈ Z. −7 −6 −5 −4 −3 −2 −1 α = 0 1 2 3 4 5 6 7 3n Any non-trivial Γ-invariant equivalence relation ≡ on Ω must be invariant under Z, which acts on labels by i → i + 1, so ≡ must be congruence mod (n) on Z for some n ≥ 2. However, Y transposes 0 and −1, and fixes 3n, so it both moves and preserves the congruence class [0] = [3n], a contradiction. Hence Γ acts primitively on Ω, so the map subgroups M are maximal in Γ. Since Z has no finite cycles on Ω, Z i has no fixed points in Ω for i = 0, so each subgroup M is non-parabolic.

One can modify this construction to give 2ℵ0 conjugacy classes of non-paraboloic maximal subgroups M of Γ by adding 1-valent vertices to an arbitrary set of free edges ‘below the horizontal axis’, as indicated by the white vertices: This changes the labelling of arcs with labels i < −3 (those below the axis), but the proof given earlier is still valid. There are 2ℵ0 choices for the set of new vertices, giving 2ℵ0 non-isomorphic maps; these give 2ℵ0 inequivalent primitive actions of Γ and hence 2ℵ0 conjugacy classes of non-parabolic maximal subgroups M.

Generalisation to other triangle groups

Theorem (J, 2018)

If p ≥ 3 and q ≥ 2 then the triangle group ∆(p, q, ∞) ∼ = Cp ∗ Cq has 2ℵ0 conjugacy classes of non-parabolic maximal subgroups. Outline proof. If q = 2 then the construction is similar to that for the modular group (where p = 3), but using p-valent planar maps. If p, q ≥ 3 a similar but more complicated construction is required, using bipartite planar maps with black and white vertices of valencies dividing p and q; in this case, the generators X and Y of

• rder p and q permute the set Ω of edges of the map, rotating

them around their incident black and white vertices. In all cases the map used has a single face, so Z has a single cycle which can be identified with Z, allowing a proof of primitivity.

The preceding proofs of primitivity depend heavily on a generator Z having infinite order. What about cocompact triangle groups ∆ = ∆(p, q, r), those with finite periods p, q and r? If p−1 + q−1 + r−1 ≥ 1 then ∆ acts on the sphere or euclidean plane, and is abelian-by-finite with at most ℵ0 maximal subgroups, all known and of finite index. We therefore restrict attention to hyperbolic triangle groups, those with p−1 + q−1 + r−1 < 1. The most interesting of these is ∆(3, 2, 7), since its finite quotients are the Hurwitz groups, those groups G attaining Hurwitz’s bound |G| ≤ 84(g − 1) for the automorphism group G of a compact Riemann surface of genus g ≥ 2. In 1980 Marston Conder showed that all but finitely many alternating groups An are Hurwitz groups, by constructing primitive permutation representations of ∆ of all sufficiently large finite degrees n. His technique can be modified to give 2ℵ0 primitive representations of ∆ of infinite degree, for a large class of cocompact hyperbolic triangle groups ∆.

Joining maps together

Conder used Graham Higman’s technique of ‘sewing together coset diagrams’ (equivalently maps), using ‘handles’. A (1)-handle in a map M representing ∆ is a pair of fixed points α, β of Y (i.e. free edges) with β = αX. For example: α β M If maps Mi (i = 1, 2) for ∆, with ni arcs, have (1)-handles αi, βi,

• ne can form a (1)-join M1(1)M2, a map for ∆ with n1 + n2 arcs,

by replacing the fixed points αi, βi of Y on Ω1 ∪ Ω2 with 2-cycles (α1, α2) and (β1, β2) (equivalently joining the free edges together), and leaving all other cycles of X and Y on Ω1 ∪ Ω2 unchanged. One can check that the defining relations of ∆ are all preserved.

Example of a (1)-join of two maps A and C, corresponding to Conder’s coset diagrams A and C, with handles shown in red. A C A(1)C A and C have monodromy groups G ∼ = PSL2(13) and PGL3(2), of degrees 14 and 21 (on points of P1(F13) and flags of P2(F2)). A(1)C has monodromy group G ∼ = A35, of degree 14 + 21 = 35.

By systematically joining coset diagrams (equivalently maps) representing ∆ = ∆(3, 2, 7), using (1)-handles and similar (2)- and (3)-handles, Conder constructed, for all n ≥ 168, permutation representations of ∆ of degree n giving epimorphisms ∆ → An. In 1981 he proved a similar result for ∆(3, 2, r) for all r ≥ 7. By joining infinitely many copies of Conder’s maps, one can obtain 2ℵ0 non-isomorphic maps representing ∆ = ∆(3, 2, r), giving 2ℵ0 inequivalent representations of ∆. As in Conder’s finite case, one can arrange that these representations are all primitive, so the point-stabilisers form 2ℵ0 conjugacy classes of maximal subgroups.

Theorem (J, 2018)

If one of p, q, r is even, another is divisible by 3 and the third is at least 7, ∆(p, q, r) has 2ℵ0 conjugacy classes of maximal subgroups. Proof Lift the maximal subgroups of ∆(3, 2, r), constructed above, back to ∆(p, q, r) via an epimorphism ∆(p, q, r) → ∆(3, 2, r).

Applications to maps

Realisation Problem Given a group A and class C of mathematical

• bjects, is A isomorphic to AutC O for some object O ∈ C ?

Theorem (Frucht, 1939)

Every finite group is isomorphic to the automorphism group of a finite graph.

Theorem (Sabidussi, 1960)

Every group is isomorphic to the automorphism group of a graph. There are similar results for many other classes of objects, e.g. Riemann surfaces, fields, hyperbolic manifolds, polytopes, etc.

Theorem (Cori and Mach` ı, 1982)

Every finite group is isomorphic to the automorphism group of a finite oriented map or hypermap. Can one extend this result?

Theorem (J, 2018)

If p ≥ 3 then given any countable group A there are 2ℵ0 non-isomorphic p-valent oriented maps M with Aut M ∼ = A. Proof p-valent oriented maps M correspond to permutation representations ∆ → G ≤ S := Sym(Ω) of ∆ = ∆(p, 2, ∞), or equivalently to conjugacy classes of subgroups M ≤ ∆. Then Aut M ∼ = CS(G) ∼ = N∆(M)/M, where C and N denote centraliser and normaliser. Therefore, to realise a group A as Aut M for such a map M it is sufficient to find a subgroup M ≤ ∆ with N∆(M)/M ∼ = A. The simplest case is when p = 3, with ∆ the modular group Γ = ∆(3, 2, ∞) = PSL2(Z).

N The map subgroups N for this map N are maximal in Γ, and are isomorphic to C2 ∗ C2 ∗ C2 ∗ C∞ ∗ C∞ ∗ · · · = C2 ∗ C2 ∗ C2 ∗ F∞ (the cyclic free factors correspond to the fixed points of Y and Z). There is an epimorphism N → F∞, and hence an epimorphism θ : N → A for every countable group A. If A = 1 there are 2ℵ0 such epimorphisms θ with kernels M not normal in Γ (so NΓ(M) = N) and mutually non-conjugate in Γ. These subgroups M correspond to 2ℵ0 non-isomorphic oriented cubic maps M with Aut M ∼ = N/M ∼ = A. Similar arguments deal with the cases A = 1 and p > 3.

A map has type {r, p} if p and r are the least common multiples

• f the valencies of its vertices and faces (Coxeter’s notation).

Theorem (J, 2018)

If p−1 + r−1 < 1/2 then given any finite group A there are ℵ0 finite oriented maps M of type {r, p} with Aut M ∼ = A. (Of course, there are only ℵ0 finite maps, up to isomorphism, so we cannot have 2ℵ0 of them here.) The proof is similar, but it uses epimorphisms ∆ := ∆(p, 2, r) → G = PSL2(n) for suitable primes n. Maximal subgroups of G lift back to maximal subgroups N of finite index in ∆. Then epimorphisms N → A realise A as N/M = N∆(M)/M for ℵ0 non-conjugate subgroups M

• f finite index in ∆, corresponding to ℵ0 non-isomorphic finite
• riented maps of type {r, p} with Aut M ∼

= A.

A new proof of Greenberg’s theorem.

In 1974 Leon Greenberg proved the following theorem.

Theorem

Every non-trivial finite group is isomorphic to the automorphism group of a compact Riemann surface. His proof depends on a delicate construction of maximal Fuchsian groups with a given signature. It should be mentioned that three dimensional version of the above theorem was obtained by Sadayoshi Kojima (1988) who proved that every finite group is isomorphic to the automorphism group of a compact hyperbolic 3-manifold. Later, it was shown by Alex Lubotzky and Misha Belolipetsky (2005) that any finite group is the full isometry group of a compact hyperbolic n-manifold for any n ≥ 4.

Here we give an alternative proof, based on well-known properties

• f triangle groups and their finite quotient groups. Given a

d-generator finite group A, choose any prime p ≥ 12d + 13, and let ∆ be the triangle group ∆ = ∆(3, 2, p) = X, Y , Z | X 3 = Y 2 = Z p = XYZ = 1

• f type (3, 2, p). The reduction mod (p) of the modular group

Γ = PSL2(Z) = ∆(3, 2, ∞) induces an epimorphism ∆ → PSL2(p), giving a primitive action of ∆ of degree p + 1 on the projective line P1(Fp). The subgroup N = ∆∞ of ∆ fixing ∞ is thus a maximal subgroup of index p + 1 in ∆. By a result of Singerman(1970) it has signature (g; 3[eX ], 2[eY ], p[eZ ]) where g is the genus of the surface H/N, and the multiplicities eX, eY and eZ

• f the periods 3, 2 and p are the numbers of fixed points of X, Y

and Z in P1(Fp).

Thus N has hyperbolic generators Ai, Bi (i = 1, . . . , g) and elliptic generators Xj (j = 1, . . . , eX), Yk (k = 1, . . . , eY ), Zl (l = 1, . . . , eZ) with defining relations

• i

[Ai, Bi] ·

• j

Xj ·

• k

Yk ·

• l

Zl = X 3

j = Y 2 k = Z p l = 1.

The Riemann–Hurwitz formula, applied to the inclusion N ≤ ∆, gives (p + 1)p − 6 6p = 2g − 2 + 2eX 3 + eY 2 + eZ

• 1 − 1

p

• .

Since ∆ acts on P1(Fp) by M¨

• bius transformations, we have

eX = 2 or 0 as p ≡ ±1 mod (3), eY = 2 or 0 as p ≡ ±1 mod (4), and eZ = 1. This implies that g = p−c

12 where c = 13, 5, 7 or −1

as p ≡ 1, 5, 7 or 11 mod (12). (By Dirichlet’s Theorem there are infinitely many primes in each of these congruence classes.)

Thus g ≥ p − 13 12 ≥ d, so there is an epimorphism θ : N → A given by mapping the elliptic generators and the hyperbolic generators Bi of N to 1, and the hyperbolic generators Ai to a set of generators for A. Let M = ker θ, so M is normal in N with N/M ∼ = A. Clearly N∆(M) ≥ N, so by the maximality of N in ∆ we must have N∆(M) = N or ∆. In the latter case M is contained in the core K

• f N in ∆, the kernel of the action of ∆ on P1(Fp). Now one can

choose this action so that ∞ is the fixed point of Z and hence Z ∈ N. The elliptic generator Z1 of N is conjugate in N to a non-identity power of Z. By the definition of θ we have Z1 ∈ M and hence Z ∈ M since Z1 and Z have the same order p. However, Z acts non-trivially on P1(Fp), so Z ∈ K and hence M ≤ K. Thus N∆(M) = N.

Since p ≥ 13, a theorem of Takeuchi (1977) implies that ∆ is non-arithmetic, so by a theorem of Margulis (1977) the commensurator Comm(∆) of ∆ is a Fuchsian group containing ∆. Now ∆ is a maximal Fuchsian group by a theorem of Singerman (1972), so Comm(∆) = ∆ and ∆ is the commensurator of each of its subgroups of finite index, including M. Thus the normaliser of M in PSL2(R) is contained in ∆, and is therefore equal to N∆(M) = N. It follows that the compact Riemann surface S = H/M has automorphism group Aut S ∼ = N/M ∼ = A, as required.

Remark 1. One cannot regard this as an elementary proof of Greenberg’s Theorem, since the results of Margulis, Singerman and Takeuchi which it uses are far from elementary. Nevertheless, the route from them to the required destination is both short and straightforward. Remark 2. It is surprisedly, but similar arguments has been used by A.D.Mednykh (1979) to give an explicit construction of Riemann surface without automorphisms.

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