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 SL 2 ( Z ) which act regularly on the primitive elements of Z 2 , those ( u , v ) ∈ Z 2 with u and v coprime. As pointed out by Magnus (1973, 1974), the images of these subgroups in the modular group Γ = PSL 2 ( Z ) = SL 2 ( Z ) / {± I } ∼ = C 3 ∗ C 2 are maximal nonparabolic subgroups, that is, maximal with respect to containing no parabolic elements. obius transformations on the upper half plane U ⊂ C (Γ acts by M¨ and on the rational projective line P 1 ( Q ) = Q ∪ {∞} . A non-identity element of Γ is parabolic if it has a fixed point in P 1 ( 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 Z α α X α Y α X 2 = orientation Then vertices, edges and faces correspond to cycles of X , Y and Z .

For any map M (cubic, oriented), the map subgroups M = Γ α = { g ∈ Γ | α g = 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 α Z i ∈ Ω labelled i ∈ Z . α = 0 1 2 3 4 6 7 3 n 5 − 1 − 2 − 3 − 5 − 6 − 4 − 7 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 3 n , so it both moves and preserves the congruence class [0] = [3 n ], 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 2 ℵ 0 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 2 ℵ 0 choices for the set of new vertices, giving 2 ℵ 0 non-isomorphic maps; these give 2 ℵ 0 inequivalent primitive actions of Γ and hence 2 ℵ 0 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 , ∞ ) ∼ = C p ∗ C q has 2 ℵ 0 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 order 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 ℵ 0 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 A n are Hurwitz groups, by constructing primitive permutation representations of ∆ of all sufficiently large finite degrees n . His technique can be modified to give 2 ℵ 0 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 = α X . For example: β α M If maps M i ( i = 1 , 2) for ∆, with n i arcs, have (1)-handles α i , β i , one can form a (1)-join M 1 (1) M 2 , a map for ∆ with n 1 + n 2 arcs, by replacing the fixed points α i , β i of Y on Ω 1 ∪ Ω 2 with 2-cycles ( α 1 , α 2 ) and ( β 1 , β 2 ) (equivalently joining the free edges together), and leaving all other cycles of X and Y on Ω 1 ∪ Ω 2 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 ∼ = PSL 2 (13) and PGL 3 (2), of degrees 14 and 21 (on points of P 1 ( F 13 ) and flags of P 2 ( F 2 )). A (1) C has monodromy group G ∼ = A 35 , 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 ∆ → A n . 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 2 ℵ 0 non-isomorphic maps representing ∆ = ∆(3 , 2 , r ), giving 2 ℵ 0 inequivalent representations of ∆. As in Conder’s finite case, one can arrange that these representations are all primitive, so the point-stabilisers form 2 ℵ 0 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 2 ℵ 0 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 objects, is A isomorphic to Aut C 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 2 ℵ 0 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 ∼ = C S ( 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 , ∞ ) = PSL 2 ( Z ) .