Maximal subgroups of triangle groups

Gareth Jones

University of Southampton, UK

Summer School for Inetnational conference and PhD-Master on Groups andGraphs, 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 ormaps on surfaces.

A great amount is known about their subgroups of finite index andtheir finite quotient groups, but less seems to be known abouttheir subgroups of infinite index and their infinite quotient groups.

I will describe some simple constructions of uncountably manyconjugacy classes of maximal subgroups (mostly of infinite index)in various hyperbolic triangle groups, generalising results ofNeumann, Magnus and others for the modular group.

The constructions have applications to the realisation of groups asautomorphism groups of maps and hypermaps on surfaces, givinganalogues of results of Frucht and Sabidussi for graphs.

In 1933 B. H. Neumann used permutations to constructuncountably many subgroups of SL2(Z) which act regularly on theprimitive elements of Z2, those (u, v) ∈ Z2 with u and v coprime.

As pointed out by Magnus (1973, 1974), the images of thesesubgroups in the modular group

Γ = PSL2(Z) = SL2(Z)/±I ∼= C3 ∗ C2

are maximal nonparabolic subgroups, that is, maximal with respectto containing no parabolic elements.

(Γ acts by Mobius transformations on the upper half plane U ⊂ Cand on the rational projective line P1(Q) = Q ∪ ∞. Anon-identity element of Γ is parabolic if it has a fixed point inP1(Q), or equivalently has trace ±2; the parabolic elements of Γare the conjugates of the powers Z i (i 6= 0) of Z : t 7→ t + 1.)

Further examples of maximal nonparabolic subgroups of Γ weresubsequently found by Tretkoff and by Brenner and Lyndon.

The modular group and cubic mapsLet M be a map (a connected graph, embedded without crossings,and with simply connected faces) on an oriented surface. Assumethat M is cubic (all vertex valencies divide 3), and allow freeedges. 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 atransitive action of Γ on Ω by letting X rotate arcs around theirincident vertices (following the orientation), and Y reverse arcs, so1-valent vertices and free edges give fixed points of X and Y .

α

αYαX

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 aremaximal if and only Γ acts primitively on Ω, i.e. preserves nonon-trivial equivalence relations on Ω.

Proposition

The map subgroups for the following map M are maximal in Γ andare non-parabolic.

(Note: ‘maximal and nonparabolic’ 6= ‘maximal nonparabolic’ !)

M

Proof. Here isM, 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 = 01

23

4 56

73n

Any non-trivial Γ-invariant equivalence relation ≡ on Ω must beinvariant under Z , which acts on labels by i 7→ i + 1, so ≡ must becongruence mod (n) on Z for some n ≥ 2. However, Y transposes0 and −1, and fixes 3n, so it both moves and preserves thecongruence class [0] = [3n], a contradiction. Hence Γ actsprimitively on Ω, so the map subgroups M are maximal in Γ. SinceZ has no finite cycles on Ω, Z i has no fixed points in Ω for i 6= 0,so each subgroup M is non-parabolic.

One can modify this construction to give 2ℵ0 conjugacy classes ofnon-paraboloic maximal subgroups M of Γ by adding 1-valentvertices 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 belowthe axis), but the proof given earlier is still valid. There are 2ℵ0

choices for the set of new vertices, giving 2ℵ0 non-isomorphicmaps; these give 2ℵ0 inequivalent primitive actions of Γ and hence2ℵ0 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 forthe 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 ofvalencies dividing p and q; in this case, the generators X and Y oforder p and q permute the set Ω of edges of the map, rotatingthem around their incident black and white vertices.

In all cases the map used has a single face, so Z has a single cyclewhich can be identified with Z, allowing a proof of primitivity.

The preceding proofs of primitivity depend heavily on a generatorZ 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 euclideanplane, and is abelian-by-finite with at most ℵ0 maximal subgroups,all known and of finite index. We therefore restrict attention tohyperbolic triangle groups, those with p−1 + q−1 + r−1 < 1.

The most interesting of these is ∆(3, 2, 7), since its finite quotientsare the Hurwitz groups, those groups G attaining Hurwitz’s bound|G | ≤ 84(g − 1) for the automorphism group G of a compactRiemann surface of genus g ≥ 2. In 1980 Marston Conder showedthat all but finitely many alternating groups An are Hurwitz groups,by constructing primitive permutation representations of ∆ of allsufficiently large finite degrees n. His technique can be modified togive 2ℵ0 primitive representations of ∆ of infinite degree, for alarge class of cocompact hyperbolic triangle groups ∆.

Joining maps togetherConder used Graham Higman’s technique of ‘sewing together cosetdiagrams’ (equivalently maps), using ‘handles’.

A (1)-handle in a map M representing ∆ is a pair of fixed pointsα, β of Y (i.e. free edges) with β = αX . For example:

αβ

M

If maps Mi (i = 1, 2) for ∆, with ni arcs, have (1)-handles αi , βi ,one 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 toConder’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), ofdegrees 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, permutationrepresentations 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 obtain2ℵ0 non-isomorphic maps representing ∆ = ∆(3, 2, r), giving 2ℵ0

inequivalent representations of ∆. As in Conder’s finite case, onecan arrange that these representations are all primitive, so thepoint-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 atleast 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 mathematicalobjects, is A isomorphic to AutCO for some object O ∈ C ?

Theorem (Frucht, 1939)

Every finite group is isomorphic to the automorphism group of afinite 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 Machı, 1982)

Every finite group is isomorphic to the automorphism group of afinite 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 AutM∼= A.

Proof p-valent oriented maps M correspond to permutationrepresentations ∆→ G ≤ S := Sym(Ω) of ∆ = ∆(p, 2,∞), orequivalently to conjugacy classes of subgroups M ≤ ∆. Then

AutM∼= CS(G ) ∼= N∆(M)/M,

where C and N denote centraliser and normaliser.

Therefore, to realise a group A as AutM for such a map M it issufficient 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 areisomorphic 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 epimorphismθ : N → A for every countable group A.

If A 6= 1 there are 2ℵ0 such epimorphisms θ with kernels M notnormal in Γ (so NΓ(M) = N) and mutually non-conjugate in Γ.

These subgroups M correspond to 2ℵ0 non-isomorphic orientedcubic maps M with AutM∼= 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 multiplesof 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 AutM∼= A.

(Of course, there are only ℵ0 finite maps, up to isomorphism, sowe 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 maximalsubgroups N of finite index in ∆. Then epimorphisms N → Arealise A as N/M = N∆(M)/M for ℵ0 non-conjugate subgroups Mof finite index in ∆, corresponding to ℵ0 non-isomorphic finiteoriented maps of type r , p with AutM∼= A.

A new proof of Greenberg’s theorem.In 1974 Leon Greenberg proved the following theorem.

TheoremEvery non-trivial finite group is isomorphic to the automorphismgroup of a compact Riemann surface.

His proof depends on a delicate construction of maximal Fuchsiangroups with a given signature.It should be mentioned that three dimensional version of the abovetheorem was obtained by Sadayoshi Kojima (1988) who provedthat every finite group is isomorphic to the automorphism group ofa 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 acompact hyperbolic n-manifold for any n ≥ 4.

Here we give an alternative proof, based on well-known propertiesof triangle groups and their finite quotient groups. Given ad-generator finite group A, choose any prime p ≥ 12d + 13, andlet ∆ be the triangle group

∆ = ∆(3, 2, p) = 〈X ,Y ,Z | X 3 = Y 2 = Zp = XYZ = 1〉

of type (3, 2, p). The reduction mod (p) of the modular groupΓ = PSL2(Z) = ∆(3, 2,∞) induces an epimorphism∆→ PSL2(p), giving a primitive action of ∆ of degree p + 1 onthe projective line P1(Fp). The subgroup N = ∆∞ of ∆ fixing ∞is thus a maximal subgroup of index p + 1 in ∆. By a result ofSingerman(1970) it has signature (g ; 3[eX ], 2[eY ], p[eZ ]) where g isthe genus of the surface H/N, and the multiplicities eX , eY and eZof the periods 3, 2 and p are the numbers of fixed points of X ,Yand Z in P1(Fp).

Thus N has hyperbolic generators Ai ,Bi (i = 1, . . . , g) and ellipticgenerators 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 3j = Y 2

k = Zpl = 1.

The Riemann–Hurwitz formula, applied to the inclusion N ≤ ∆,gives

(p + 1)p − 6

6p= 2g − 2 +

2eX3

+eY2

+ eZ

(1− 1

p

).

Since ∆ acts on P1(Fp) by Mobius transformations, we haveeX = 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 −1as p ≡ 1, 5, 7 or 11 mod (12). (By Dirichlet’s Theorem there areinfinitely 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 theelliptic generators and the hyperbolic generators Bi of N to 1, andthe hyperbolic generators Ai to a set of generators for A. LetM = ker θ, so M is normal in N with N/M ∼= A. ClearlyN∆(M) ≥ N, so by the maximality of N in ∆ we must haveN∆(M) = N or ∆. In the latter case M is contained in the core Kof N in ∆, the kernel of the action of ∆ on P1(Fp). Now one canchoose this action so that ∞ is the fixed point of Z and henceZ ∈ N. The elliptic generator Z1 of N is conjugate in N to anon-identity power of Z . By the definition of θ we have Z1 ∈ Mand hence Z ∈ M since Z1 and Z have the same order p.However, Z acts non-trivially on P1(Fp), so Z 6∈ K and henceM 6≤ K . Thus N∆(M) = N.

Since p ≥ 13, a theorem of Takeuchi (1977) implies that ∆ isnon-arithmetic, so by a theorem of Margulis (1977) thecommensurator 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 ofits subgroups of finite index, including M. Thus the normaliser ofM in PSL2(R) is contained in ∆, and is therefore equal toN∆(M) = N. It follows that the compact Riemann surfaceS = H/M has automorphism group AutS ∼= N/M ∼= A, asrequired.

Remark 1. One cannot regard this as an elementary proof ofGreenberg’s Theorem, since the results of Margulis, Singerman andTakeuchi which it uses are far from elementary. Nevertheless, theroute from them to the required destination is both short andstraightforward.

Remark 2. It is surprisedly, but similar arguments has been usedby A.D.Mednykh (1979) to give an explicit construction ofRiemann surface without automorphisms.

THANK YOU FOR LISTENING!