Journal of Pure and Applied Algebra 218 (2014) 777–783

Contents lists available at ScienceDirect

Journal of Pure and Applied Algebra

www.elsevier.com/locate/jpaa

Generating pairs and group actions

Darryl McCullough 1

Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 December 2012Received in revised form 4 September 2013Available online 7 November 2013Communicated by C.A. Weibel

MSC:Primary: 57M60; secondary: 20F99

An action of a finite group on a closed 2-manifold is called almost free if it has a singleorbit of points with nontrivial stabilizers. It is called large when the order of the group isgreater than or equal to the genus of the surface. We prove that the orientation-preservinglarge almost free actions of G on closed orientable surfaces correspond to the Nielsenequivalence classes of generating pairs of G . We classify the almost free actions on thesurfaces of genera 3 and 4, find the large almost free actions of the alternating group A5,and give various other examples.

© 2013 Elsevier B.V. All rights reserved.

For a two-generator group, two generating pairs are called Nielsen equivalent if one can be obtained from the otherby a sequence of operations of replacing one generator by one of its two products with the other or with the inverse ofthe other. More generally, one can consider a similar relation on generating n-tuples. The equivalence classes so obtainedhave been examined or used both in algebraic contexts [5,7–9,12,28,26,30,33,35] and in topological ones [16–21,24,27]. Inthis paper we will establish a new topological interpretation of Nielsen classes of generating pairs in terms of finite groupactions on closed orientable surfaces.

As usual, an action of a group G on a compact surface Σ is an injective homomorphism ϕ : G → Diff(Σ). We willassume without mention that all actions preserve orientation. Two actions ϕi : G → Diff(Σi), i = 1,2, are called equivalentif they are the same up to an identification of Σ1 with Σ2, that is, if there is a diffeomorphism h : Σ1 → Σ2 such thatϕ2(g) = hϕ1(g)h−1 for all g ∈ G . An action of a finite group G is called almost free if

⋃1 �=g∈G Fix(g) is nonempty and G acts

transitively on it. Almost free actions may be regarded as the non-free actions that are as close as possible to being free. Forclosed 2-manifolds, they correspond exactly to the free actions on bounded surfaces which act transitively with nontrivialstabilizers on the set of boundary components.

We call an almost free action of G on a closed, orientable surface Σ of genus g large when |G| � g . On the face of it, thisconflicts with the usage of the term large introduced by R. Kulkarni [15], to wit, that |G| > 4(g − 1). However, we will see inProposition 2.1(a) that for an almost free action, |G| � 4(g − 1), so almost free actions are never large in the sense of [15].Moreover, our use of the term is in the same spirit as Kulkarni’s, so we are willing to request the reader’s indulgence. Weremark also that no almost free action determines a branched covering over the sphere, as the latter actions all have atleast two orbits of fixed points.

It will be evident that, in the language of J.W. Anderson and A. Wootton [1], almost free actions are exactly the actionshaving skeletal signature of the form (h,1), and large almost free actions are those of skeletal signature (1,1).

Our main result shows that large almost free actions are classified by the Nielsen classes of generating pairs of G .

Main Theorem. A group G admits a large almost free action on some closed orientable surface if and only if G is a nonabeliantwo-generator group. The equivalence classes of all such actions of G on closed orientable surfaces correspond bijectively to the Nielsenclasses of generating pairs of G.

1 This work was supported in part by NSF grant DMS-0802424.

0022-4049/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jpaa.2013.10.001

778 D. McCullough / Journal of Pure and Applied Algebra 218 (2014) 777–783

The Main Theorem follows directly from Theorem 2.3 and its auxiliary results in Section 2. Additional information aboutthe actions is obtained; for example, if g is the genus of the surface, then 2(g − 1) < |G| � 4(g − 1). Theorem 2.3 alsoprovides a version of the Main Theorem for weak equivalence classes of G-actions, defined below.

Anderson and Wootton [1, Theorem 3.8] showed that infinitely many genera of surfaces admit actions of skeletal signa-ture (1,1), and infinitely many genera do not. In Section 3, we will see that every odd genus at least 3 admits two largealmost free actions of dihedral groups with skeletal signature (1,1), and that for each N , there are a genus g and a primep such that the surface of genus g admits at least N (weak) equivalence classes of large almost free actions of SL(2, p).

Section 1 reviews Nielsen equivalence and T -equivalence, and Section 2 develops the above-mentioned results that implythe Main Theorem. In Section 3, in addition to the results mentioned above, we find the almost free actions on the surfacesof genera 3 and 4, and the large almost free actions of the alternating group A5. We also give other examples, such as alarge almost free action that is a minimal-genus action. Topological description of the actions treated in the Main Theoremis discussed in Section 4.

The author is grateful to the referee and to S.A. Broughton for helpful comments on the manuscript.

1. Nielsen equivalence

Throughout we will denote the free group on two generators a and b by F2. The automorphism group Aut(F2) is gener-ated by the three involutions defined by

(1) r(a) = a−1, r(b) = b,(2) s(a) = b, s(b) = a,(3) t(a) = a−1, t(b) = ab.

To verify this, one may check that Nielsen’s standard generators of Aut(F2) [29] can be written as compositions of theseinvolutions.

Let K be a two-generator group, and denote by G2(K ) the generating pairs of K , that is, the ordered pairs (A, B) such that{A, B} generates K . If we associate to (A, B) the homomorphism sending a to A and b to B , then these correspond to thesurjective homomorphisms Epi(F2, K ) from F2 to K . There is a left Aut(F2)-action on Epi(F2, K ) defined by φ ·ρ = ρ ◦ φ−1.The corresponding actions of r, s, and t on pairs in G2(K ) are

(1) r(A, B) = (A−1, B),(2) s(A, B) = (B, A),(3) t(A, B) = (A−1, AB).

We define two generating pairs in G2(K ) to be Nielsen equivalent when they lie in the same orbit for this Aut(F2)-action.More concretely, two pairs are Nielsen equivalent if and only if each can be obtained from the other by a sequence ofoperations of replacing an element by its inverse, or pre- or post-multiplying one element by the other or the inverse ofthe other, or by interchanging the two elements, since these are the effects of the standard generators of Aut(F2).

A Nielsen equivalence class of generating pairs is called a Nielsen class. Now, extend the action of Aut(F2) on Epi(F2, K )

to an action of Aut(K ) × Aut(F2) by letting (α,φ) · ρ = α ◦ ρ ◦ φ−1, or at the level of pairs, α(A, B) = (α(A),α(B)). Theresulting equivalence classes are called T -systems.

2. Almost free actions

In this section, we will prove our main technical result, Theorem 2.3. The basic idea of the proof is that large almost freeactions are encoded as equivalence classes of extensions

1 → π1(Σ) → Γ → G → 1,

where Γ has presentation 〈A, B | [A, B]n = 1〉 for some n � 2, and it turns out that two such extensions are equivalentexactly when the image pairs in G of the standard generating pair (A, B) of Γ are Nielsen equivalent.

Fix an almost free action of a finite group G on a surface Σ of genus g . For almost free actions, the general descriptionof actions in terms of signature (see for example [3]) simplifies considerably. There is only one branch point for Σ → Σ/G ,so the signature is a pair (g0;n), where g0 is the topological genus of Σ/G , and the order n of the unique branch pointsatisfies the Riemann–Hurwitz formula

2g − 2 = |G|(2g0 − 1 − 1/n).

There is an associated extension

1 → π1(Σ) → Γp−→ G → 1 (1)

where Γ is a Fuchsian group with presentation

D. McCullough / Journal of Pure and Applied Algebra 218 (2014) 777–783 779

⟨A1, B1, . . . , Ag0 , B g0 , C

∣∣∣( g0∏

i=1

[Ai, Bi])

C = Cn = 1

⟩.

We now make a sequence of observations that will prepare us for the proof of Theorem 2.3.Since π1(Σ) is torsion-free, the image of the element C in G has order n. Therefore the image of the element∏g0

i=1[Ai, Bi] also has order n, so G is nonabelian. This observation appeared in the proof of Theorem 3.8 of [1].From the Riemann–Hurwitz formula, we have

g0 = g − 1

|G| + n + 1

2n.

Since n > 1, this implies that |G| �= g − 1 and |G| � 4(g − 1). It also shows that g � 2, but in fact it shows that g � 3. For ifg = 2, we must have g0 = 1, and then |G| = 2n

n−1 � 4 and G would be abelian.

If 2 � g0 and we find that |G| < 45 (g − 1), in particular, the action is not large. When g0 = 1, we have

g = 1 + n − 1

2n|G|.

It follows that 2(g − 1) < |G|, and the action is large. Since g0 = 1 and there is a single branch point, large almost freeactions are exactly the actions said to have skeletal signature (1,1) in the terminology of Anderson and Wootton [1].

We collect the previous observations as a proposition.

Proposition 2.1. If G admits an almost free action on a closed orientable surface of genus g, then G is nonabelian and 3 � g. Moreover,

(a) |G| � 4(g − 1), and |G| �= g − 1.(b) If the action is not large, then |G| < 4

5 (g − 1).

(c) If the action is large, then the quotient space is topologically the torus, and 2(g − 1) < |G| and g = 1 + n−12n |G|.

Part (b) of Proposition 2.1 gives a quick estimate:

Corollary 2.2. If a closed surface admits an almost free action that is not large, then its genus is at least 9. If the acting group is not D3 ,then the genus is at least 11.

In Subsection 3.3, we will see that this estimate is sharp.Actions ϕi : G → Diff(Σ) are weakly equivalent if they are equivalent up to an automorphism of G , that is, if there are a

diffeomorphism h : Σ → Σ and an automorphism α : G → G so that ϕ2(g) = hϕ1(α(g))h−1 for all g ∈ G . In the literatureof group actions, this condition is often used as the definition of equivalence. Those who prefer to retain that terminologymay read “strongly equivalent” and “equivalent” respectively in place of our “equivalent” and “weakly equivalent”.

The G-actions on Σ may be encoded by epimorphisms p : Γ → G . According to [3], the weak equivalence classes cor-respond to the orbits of the natural action of the group Aut(Γ ) × Aut(G). Tracing through the definitions shows that theequivalence classes correspond to the orbits of the action of Aut(Γ ) alone.

To clarify this, an element (φ,α) of Aut(Γ ) × Aut(G) determines an isomorphism of extensions

1 −−−−→ π1(Σ1) −−−−→ Γp−−−−→ G −−−−→ 1⏐⏐� ⏐⏐�φ|π1(Σ1)

⏐⏐�φ

⏐⏐�α

⏐⏐�1 −−−−→ π1(Σ2) −−−−→ Γ

α◦p◦φ−1

−−−−−→ G −−−−→ 1

.

As detailed in [3], which refers to earlier work of W. Harvey [13] and A.M. Macbeath [23], this isomorphism correspondsto a weak equivalence h : Σ1 → Σ2 such that ϕ2(g) = hϕ1(α(g))h−1. In particular, the weak equivalence is actually anequivalence exactly when α is the identity map of G .

As in Section 1, G2(G) will denote the set of generating pairs of G . To define T -systems there, we used the action ofAut(G) on G2(G) given by α(A, B) = (α(A),α(B)). If α : G → H is a surjective homomorphism, then there is an inducedfunction from G2(G) to G2(H) defined by the same formula. Note that if (A, B) is Nielsen equivalent to (A′, B ′), then α(A, B)

is Nielsen equivalent to α(A′, B ′).

Theorem 2.3. Let G be a nonabelian finite group. Then the equivalence classes (respectively, weak equivalence classes) of large almostfree actions of G on closed orientable surfaces correspond bijectively to the Nielsen classes (respectively, T -systems) of G2(G). For anaction on a surface Σ corresponding to the Nielsen class of a generating pair (A, B), the order of the stabilizers of the fixed pointsequals the order n of [A, B] in G. Moreover, n = |G| and the genus of Σ is 1 + n−1 |G|.

|G|+χ(Σ) 2n

780 D. McCullough / Journal of Pure and Applied Algebra 218 (2014) 777–783

In particular, if G cannot be generated by only two elements then G does not admit a large almost free action on any Σ .From now on, we write Γn for the group 〈A, B | [A, B]n = 1〉. Notice that this is simply the group 〈A, B, C | [A, B]C =

Cn = 1〉 in Eq. (1) with the generator C eliminated.The next lemma is Theorem 1(2.1) of [33]. We remark that [33] is a careful correction of earlier work, in particular of

the paper [31,32], where the lemma appeared as Lemma 3. See also [11].

Lemma 2.4. Any two generating pairs of Γn are Nielsen equivalent.

Proof of Theorem 2.3. As we have seen, a large almost free action ϕ : G → Diff(Σ) gives an extension

1 → π1(Σ) → Γnp−→ G → 1.

We associate to ϕ the generating pair

P (ϕ) = p(A, B) = (a,b) ∈ G2(G).

By Lemma 2.4, the Nielsen class of P (ϕ) does not depend on the choice of generating pair (A, B) of Γn .In the remainder of the proof, we will treat the cases of equivalence and weak equivalence simultaneously, with the

parenthetical words to be used for the latter case.We first show that the generating pairs of G associated to (weakly) equivalent actions are Nielsen equivalent

(T -equivalent). If p(A, B) is one pair, the other can be written as α ◦ p ◦ φ−1(A, B), where α is the identity (an iso-morphism) of G . By Lemma 2.4, (A, B) is Nielsen equivalent to φ−1(A, B), so p(A, B) is Nielsen equivalent (T -equivalent)to α ◦ p ◦ φ−1(A, B).

Next we show that P is injective as a function from (weak) equivalence classes of actions to Nielsen classes (T -systems)in G2(G). Fix two actions ϕi : G → Diff(Σi) that are assigned the same Nielsen class (T -system), that is, for which p2(A, B)

is Nielsen equivalent (T -equivalent) to αp1(A, B), where α is the identity (is an automorphism of G).Let q : F2 → Γn be the epimorphism sending (a,b) to (A, B). Since the pairs αp1q(a,b) and p2q(a,b) are Nielsen

equivalent (T -equivalent), there is an automorphism Φ of F2 such that αp1q = p2qΦ . The generating automorphismsr, s, and t of F2 induce automorphisms of Γn , so there is an automorphism φ of Γn such that φq = qΦ . Thereforeαp1φ

−1(A, B) = p2(A, B), that is, the actions are (weakly) equivalent.Finally, a generating pair (A, B) of G for which [A, B] has order n defines a surjective homomorphism Γn → G , which is

injective on the cyclic subgroup generated by [A, B]. In a one-relator group, every torsion element is conjugate to a power ofthe relator (see for example Theorem IV.5.2 of [22]). Therefore the kernel of the homomorphism is torsion-free, so there isan action corresponding to this extension. Since G is nonabelian, n � 2 and there is a branch point. Since Γn acts transitivelyon the fixed points of its nonzero elements, the action is almost free. The Euler characteristic calculations shown above givethe genus of Σ to be 1 + n−1

2n |G|, so the almost free action is large, and the formula for n follows. �3. Examples

3.1. Dihedral groups

For any dihedral group Dk , k � 2, there is a unique Nielsen class of generating pair. This is easily checked directly (seeTheorem 14 of [24]). By Theorem 2.3, there is a unique large almost free action of Dk for each k � 3. For odd k, [A, B] hasorder k, giving g = k, while for even k, [A, B] has order k/2, giving g = k − 1.

3.2. Special linear groups

The special and projective special linear groups SL(2,q) and PSL(2,q) provide interesting examples of large almost freeactions. Information about Nielsen equivalence in the groups SL(2,q) and PSL(2,q) was obtained in [25] and [26]. As wewill now discuss, the trace plays an important role.

A well-known observation of D. Higman is that for any two-generator group, the union of the conjugacy classes of [A, B]and [B, A] is an invariant of the Nielsen class of a generating pair (A, B). Consequently, the trace of [A, B] is an invariantof the Nielsen class of a generating pair of SL(2, p). Moreover, by a result due to Schreier and van der Waerden [34] (seealso [4,6], and the appendix to [14]), the automorphisms of SL(2, p) are all induced by conjugation by matrices in GL(2, p).Therefore the trace of the commutator is an invariant of the T -system of a generating pair.

Using information about T -systems in SL(2, p), we can check that a surface may admit an arbitrarily large number oflarge almost free actions of a single group.

Proposition 3.1. Let N be a positive integer. Then for some prime p, the surface of genus 1 + (p + 1)p(p − 2)/2 admits at least Nweak equivalence classes of large almost free actions of SL(2, p).

D. McCullough / Journal of Pure and Applied Algebra 218 (2014) 777–783 781

Proof. Consider a prime p � 13. The coefficient field Fp contains ϕ(p − 1)/2 pairs {β,1/β} of elements of multiplicative

order p − 1, where ϕ is the Euler totient function. The corresponding matrices(

β 00 1/β

)have order p − 1, and have distinct

traces since β and 1/β are the roots of x2 − (β + 1/β)x + 1. Since p � 13 and β �= ±1, Theorem 2.2 of [25] shows that(β 00 1/β

)is the commutator of a generating pair of SL(2, p). As noted above, the generating pairs so obtained represent

distinct T -systems of generating pairs, since the traces of their commutators are distinct. Since the commutators have orderp − 1, Theorem 2.3 shows that they correspond to distinct weak equivalence classes of large almost free actions on thesurface of genus

g = 1 + p − 2

2(p − 1)

∣∣SL(2, p)∣∣ = 1 + p − 2

2(p − 1)(p + 1)p(p − 1) = 1 + (p + 1)p(p − 2)/2.

Choosing p with ϕ(p − 1) � 2N gives our result. �3.3. Genus 3 almost free actions

Using Theorem 2.3, we can quickly enumerate the almost free actions on the surface of genus 3. Proposition 2.1 showsthat G is nonabelian and |G| � 8, so G must be one of D3, D4, or Q 8. By Corollary 2.2, any such action is large. We havealready seen the actions for D3 and D4. For Q 8 there is a generating pair with [A, B] of order 2, easily checked to be uniqueup to Nielsen equivalence, and Theorem 2.3 applies.

We remark that the classification in Table 5 of [3] confirms this enumeration. In Section 4, we will show how to describethese actions explicitly.

The almost free action of D3 on the surface of genus 3 shows that Corollary 2.2 is sharp, since adding six handlesequivariantly near a generic orbit of this action produces an almost free action of D3 on the surface of genus 9. Similarly,almost free actions on the surface of genus 11 can be obtained from the actions of D4 and Q 8.

3.4. Genus 4 almost free actions

For g = 4, Proposition 2.1 gives 6 < |G| � 12, so G must be one of D4, D5, D6, Q 8, A4, or the semidirect product Z3 �Z4.We have already seen that the large almost free actions of D4 and Q 8 are on the surface of genus 3, while for D5 and D6the actions are in genus 5.

For A4, we regard A4 as PSL(2,3). Note that the commutator of elements of PSL(2,q) is a well-defined matrix, andconsequently its trace is well-defined. By Theorem 2.1 of [25], the only possible trace for the commutator of a generatingpair (A, B) of PSL(2,3) is 0 ∈ F3. Then, [A, B]2 = −I is the identity element, so n = 2. This gives g = 1 + |A4|/4 = 4, sothere is an almost free action of A4 on the surface of genus 4. Hand calculations or brute-force computation show that thistrace exactly classifies the Nielsen classes of G2(PSL(2,3)) (see [26]), so there is a unique Nielsen class and hence a uniquealmost free action up to equivalence.

For the semidirect product Z3 �Z4, a generating pair must contain an element of order 4. It is easy to check that thereis a unique Nielsen class of generating pairs, and that the commutator of any two generators has order n = 3, giving g = 5.We conclude that the action of A4 is the unique almost free action on the surface of genus 4. This is confirmed by theclassification of genus 4 actions by O. Bogopol’skiı̆ [2].

3.5. The alternating group A5

Among the special linear groups is SL(2,4), which is isomorphic to A5. By Theorem 2.2 of [25], the traces of the com-mutators of generating pairs of SL(2,4) are the nonzero elements {1, u, u2} of F4, where we write u for the generator ofF4 called Z(4) by the GAP system for computational algebra [10]. As in [26], hand calculations or brute-force computationshow that this trace exactly classifies the Nielsen classes of G2(SL(2,4)). Using Theorem 2.3 and elementary calculation, wefind

(1) The element 1 is the trace of matrices of order n = 3, so the Nielsen class with trace invariant 1 corresponds to analmost free action of A5 on the surface of genus g = 1 + 3−1

2·3 · 60 = 21.(2) The elements u and u2 are the traces of matrices of order n = 5 (if v is the generator of F16 called Z(16) by GAP,

then these elements of SL(2,4) are conjugate in SL(2,16) to the diagonal matrices with entries v and v−1 for the traceu2 = v + v−1, and to the diagonal matrices with entries v2 and v−2 for the trace u = v2 + v−2). These correspond totwo equivalence classes of large almost free actions on the surface of genus g = 1 + 5−1

2·5 · 60 = 25. These actions areweakly equivalent since u and u2 are related by the Frobenius automorphism of F4, which induces an automorphismon SL(2,4).

782 D. McCullough / Journal of Pure and Applied Algebra 218 (2014) 777–783

Fig. 1. The Cayley graph of Q 8 and its thickening to an orientable surface.

3.6. An almost free minimal-genus action

One might ask whether a minimal-genus action of a group can be almost free. As we will now check, one family of suchgroups are those of the form

G = ⟨A, B, C

∣∣ A2p = B2p = C2 = [A, C] = [B, C] = 1, [A, B] = C⟩,

where p is prime and p � 7. This group is a semidirect product (C2p × C2) � C2p , where the generator A of the quotientC2p acts by AB A−1 = BC , and also is a central extension

1 → C2 → G → C2p × C2p → 1.

For the generating pair (A, B), the commutator [A, B] = C has order 2, so Theorem 2.3 shows that G has a large almost freeaction on the surface of genus 2p2 + 1.

We claim that this action achieves the minimal genus of any action of G . Consider first an action which determines abranched covering over the 2-sphere with three branch points with branching orders b1, b2, and b3. As in (1.2) of [3], theaction determines a generating vector (g1, g2, g3) of G for which the order of gi is bi and g1 g2 g3 = 1.

The subgroup K of G generated by Ap and B p is dihedral of order 8 and normal in G . Suppose for contradiction thatsome gi , say g3, lies in K . The quotient G/K is C p × C p , and the images of g1 and g2 must generate G/K . But this isimpossible since g1 g2 ∈ K . Every element of G not in K has order at least p, so all bi � p. The Riemann–Hurwitz formulathen gives

g = 1 + 1

2|G|

(1 − 1

b1− 1

b2− 1

b3

)� 1 + 4p(p − 3).

Since p � 7, this exceeds 2p2 + 1. The case of the 2-sphere with four branch points is easily ruled out by similar consider-ations; the complicated term in the above calculation becomes 2 − 1/b1 − 1/b2 − 1/b3 − 1/b4 with at least three bi � p. Allother cases are immediately ruled out just using the Riemann–Hurwitz formula.

4. Geometric realization

To describe large almost free actions geometrically, we start with the extension

1 → K → F2 → G → 1,

where the homomorphism from F2 to G sends a to A and b to B for a generating pair (A, B) representing a given Nielsenclass. The two-leaved rose with fundamental group generators a and b corresponding to the two leaves admits a |G|-foldcovering space corresponding to the kernel K , in fact this covering is exactly the Cayley graph of G with respect to thegenerating set {A, B}. We can thicken this to an orientable surface Σ0 with |G|/n boundary components, where n is theorder of [A, B] in G , so that

(1) The action of G on the Cayley graph extends to an action on Σ0 that preserves the two sides.(2) The boundary circles of Σ0 represent conjugates of [A, B]n .

Indeed, the action on Σ0 exists because it is just the covering space of the once-punctured torus corresponding to the aboveextension. Attaching disks to these boundary circles, the left action of G on the covering graph extends to give the desiredaction on a closed surface Σ , branched over a torus with a single branch point.

D. McCullough / Journal of Pure and Applied Algebra 218 (2014) 777–783 783

Fig. 1 illustrates this process for the almost free action of the quaternion group Q 8 = 〈a,b | a2 = b2, bab−1 = a−1〉 onthe surface of genus 3. On the left is the Cayley graph, with edges corresponding to a and b denoted by single and doublearrows respectively. On the right is the equivariant thickening Σ0 with four boundary circles representing conjugates of[A, B]2. Attaching disks to these, and extending the action, the center points of the four disks form the unique orbit offixed points. The other two almost free actions on the surface of genus 3 can be obtained by such a construction, of course,but can also be described as obtained from the standard D3-action on S2 and an action of D4 on the torus by attaching1-handles.

References

[1] J.W. Anderson, A. Wootton, A lower bound for the number of group actions on a compact Riemann surface, Algebr. Geom. Topol. 12 (2012) 19–35.[2] O.V. Bogopol’skiı̆, Classifying the actions of finite groups on orientable surfaces of genus 4, Siberian Adv. Math. 7 (1997) 9–38; translation of Proceed-

ings of the Institute of Mathematics, vol. 30, Izdat. Ross. Akad. Nauk, Sibirsk. Otdel. Inst. Mat., Novosibirsk, 1996, pp. 48–69 (in Russian).[3] S.A. Broughton, Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69 (1991) 233–270.[4] R.W. Carter, Simple Groups of Lie Type, Wiley Class. Lib., John Wiley & Sons Inc., New York, 1989, reprint of the 1972 original, Wiley–Interscience

Publication.[5] D.J. Collins, Generation and presentation of one-relator groups with centre, Math. Z. 157 (1977) 63–77.[6] J. Dieudonné, On the Automorphisms of the Classical Groups, Mem. Am. Math. Soc., vol. 1951, 1951, vi+122 pp., with a supplement by Loo-Keng Hua.[7] M.J. Dunwoody, On T -systems of groups, J. Aust. Math. Soc. 3 (1963) 172–179.[8] M.J. Dunwoody, Nielsen transformations, in: Computational Problems in Abstract Algebra, Proc. Conf., Oxford, 1967, Pergamon, Oxford, 1970, pp. 45–46.[9] M.J. Evans, T -systems of certain finite simple groups, Math. Proc. Camb. Philos. Soc. 113 (1993) 9–22.

[10] The GAP Group, GAP—Groups, algorithms, and programming, version 4.6.5, 2013.[11] J. Gilman, Two-Generator Discrete Subgroups of PSL(2,R), Mem. Am. Math. Soc., vol. 117, 1995, x+204 pp.[12] R. Gilman, Finite quotients of the automorphism group of a free group, Can. J. Math. 29 (1977) 541–551.[13] W.J. Harvey, On branch loci in Teichmüller space, Trans. Am. Math. Soc. 153 (1971) 387–399.[14] L.K. Hua, On the automorphisms of the symplectic group over any field, Ann. Math. (2) 49 (1948) 739–759.[15] R.S. Kulkarni, Riemann surfaces admitting large automorphism groups, in: Extremal Riemann Surfaces, San Francisco, CA, 1995, in: Contemp. Math.,

vol. 201, Amer. Math. Soc., Providence, RI, 1997, pp. 63–79.[16] M. Lustig, Nielsen equivalence and simple-homotopy type, Proc. Lond. Math. Soc. (3) 62 (1991) 537–562.[17] M. Lustig, Y. Moriah, Nielsen equivalence in Fuchsian groups and Seifert fibered spaces, Topology 30 (1991) 191–204.[18] M. Lustig, Y. Moriah, Generalized Montesinos knots, tunnels and N -torsion, Math. Ann. 295 (1993) 167–189.[19] M. Lustig, Y. Moriah, Generating systems of groups and Reidemeister–Whitehead torsion, J. Algebra 157 (1993) 170–198.[20] M. Lustig, Y. Moriah, N -torsion and applications, in: Geometric Group Theory, vol. 1, Sussex, 1991, in: Lond. Math. Soc. Lect. Note Ser., vol. 181,

Cambridge Univ. Press, Cambridge, 1993, pp. 159–168.[21] M. Lustig, Y. Moriah, On the complexity of the Heegaard structure of hyperbolic 3-manifolds, Math. Z. 226 (1997) 349–358.[22] R.C. Lyndon, P.E. Schupp, Combinatorial Group Theory, Class. Math., Springer-Verlag, Berlin, 2001, reprint of the 1977 edition.[23] A.M. Macbeath, The classification of non-euclidean plane crystallographic groups, Can. J. Math. 19 (1967) 1192–1205.[24] D. McCullough, M. Wanderley, Free actions on handlebodies, J. Pure Appl. Algebra 181 (2003) 85–104.[25] D. McCullough, M. Wanderley, Writing elements of PSL(2,q) as commutators, Commun. Algebra 39 (2011) 1234–1241.[26] D. McCullough, M. Wanderley, Nielsen equivalence of generating pairs of SL(2,q), Glasg. Math. J. 55 (2013) 481–509.[27] Y. Moriah, Heegaard splittings of Seifert fibered spaces, Invent. Math. 91 (1988) 465–481.[28] B.H. Neumann, H. Neumann, Zwei Klassen charakteristischer Untergruppen und ihre Faktorgruppen, Math. Nachr. 4 (1951) 106–125.[29] J. Nielsen, Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 78 (1917) 385–397.[30] S.J. Pride, The isomorphism problem for two-generator one-relator groups with torsion is solvable, Trans. Am. Math. Soc. 227 (1977) 109–139.[31] N. Purzitsky, G. Rosenberger, Two generator Fuchsian groups of genus one, Math. Z. 128 (1972) 245–251.[32] N. Purzitsky, G. Rosenberger, Correction to: “Two generator Fuchsian groups of genus one” (Math. Z. 128 (1972) 245–251), Math. Z. 132 (1973) 261–262.[33] G. Rosenberger, All generating pairs of all two-generator Fuchsian groups, Arch. Math. 46 (1986) 198–204.[34] O. Schreier, B.L. van der Waerden, Die Automorphismen der projektiven Gruppen, Abh. Math. Semin. Univ. Hamb. 6 (1928) 303–322.[35] P.J. Webb, Minimal relation modules of free nilpotent groups, Arch. Math. 37 (1981) 193–197.