1 Introduction

Regular maps are cellular decompositions of closed surfaces exhibiting the largest possible number of symmetries, or the largest number of these that preserve orientation. The study of regular maps grew out of work in the late 1800s by Dyck and Burnside and others on group actions on surfaces, and was progressed substantially by Brahana [1]. Since those earliest times, regular maps have been closely related to their automorphism groups.

The main reason behind this is the fact that the group of all orientation-preserving automorphisms of any regular map on an orientable surface having p-gonal faces, with q faces meeting at each vertex, is a smooth finite quotient of the ordinary triangle group

$$\begin{aligned} \Delta (q,2,p) = \langle \, a,b \mid a^q = b^2 = (ab)^p = 1 \,\rangle , \end{aligned}$$

with ‘smooth’ (here) meaning that the orders of the elements a, b and ab are preserved, or equivalently, that the kernel is torsion-free. Conversely, any smooth homomorphism from \(\Delta (q,2,p)\) to a finite group G gives rise to a unique regular map with p-gonal faces, q at each vertex, whose orientation-preserving automorphism group is isomorphic to G.

In this paper we address the problem of classification of regular maps with a given automorphism group. This problem has received relatively little attention in comparison to classifying regular maps on a given surface, or with a given underlying graph. Complete classification results are very rare and, aside from the folklore result describing regular maps with abelian automorphism groups, most of them are related to finite simple groups—see [11, 12, 19], for example.

Here we take another viewpoint, and concentrate our attention on orientably-regular maps for which the orientation-preserving automorphism group is nilpotent. We will call these nilpotent maps. Our attention is restricted to regular maps on orientable surfaces, because the only non-orientable regular maps with nilpotent automorphism group are cycles of length \(2^n\) (for each \(n\ge 1\)) embedded in the projective plane, and their geometric duals; see [22, p. 450] or [9, Theorem 3.4].

In contrast to the abelian case, the class of nilpotent maps is rich. This class includes, for example, regular embeddings of the complete bipartite graphs \(K_{n,n}\) and regular embeddings of the n-dimensional cubes, when n is a power of 2; indeed, the automorphism groups of all such maps are 2-groups (since the numbers of vertices and the orders of vertex-stabilisers are powers of 2). Recently completed classifications of these two families of maps (in [3, 6–8]) show that both complete bipartite maps and n-cube maps exhibit a great variety of forms, and suggest that achieving a complete classification of nilpotent regular maps may be a very difficult problem.

The first thorough study of nilpotent regular maps was undertaken by Malnič, Nedela and Škoviera in [16]. Two fundamental observations were made in the latter study: first, a nilpotent regular map with non-abelian automorphism group must be bipartite (see [16, Theorem 3.4]), and second, every nilpotent regular map can be uniquely decomposed into a direct product of a regular map whose automorphism is a 2-group and a spherical regular map consisting of a single vertex and odd number of semi-edges (see [16, Theorem 3.2]). Moreover, it is easy to see that the second factor adds only edges that are parallel to the edges of the first factor. It follows that nilpotent maps tend to have multiple edges, and those with simple underlying graph must arise from 2-groups.

It is also known that every orientably-regular map \(\mathcal {M}\) with multiple edges gives rise to two smaller regular maps: a quotient regular map \(\mathcal {M}'\) which has the same number of vertices as \(\mathcal {M}\) but with simple underlying graph, and a regular embedding \(\mathcal {M}''\) of the dipole graph which has two vertices and the same multiplicity as \(\mathcal {M}\); see [23, Theorem 3] or [9, Theorem 2.1]. Moreover, if \(\mathcal {M}\) is nilpotent, then both \(\mathcal {M}'\) and \(\mathcal {M}''\) are also nilpotent. It follows that the problem of classifying nilpotent maps splits into the problem of classifying nilpotent maps with simple underlying graph, the problem of classifying nilpotent dipoles, and the problem of determining how these two types of regular maps may be combined.

As shown in [16], nilpotent dipoles are not difficult to classify. In fact, for every integer \(\mu \ge 2\), up to isomorphism there are only one, two, or four nilpotent dipole maps with edge-multiplicity \(\mu \); see [16, Theorem 4.6]. Hence the main part of our classification problem involves understanding nilpotent maps with simple underlying graph.

Our main objective in this paper is to show that for any positive integer c, there are only finitely many regular maps with simple underlying graph, such that the orientation-preserving automorphism group of the map is nilpotent of class c. In fact, we prove that the number of vertices of any such map is bounded by a function of c. Also we give an exact formula for the maximum number of vertices of a simple nilpotent map of class c, and show that this maximum is achieved by exactly one nilpotent simple map \(\mathcal {M}_c\), and that every other simple nilpotent map of the given class c is a quotient of \(\mathcal {M}_c\).

We do this in Sect. 3, after giving some further background in Sect. 2, and to complete the paper we give all nilpotent simple maps of class 2 to 4 in Sect. 4.

2 Preliminaries

In this section we provide some further background, first on orientably-regular maps, and second on nilpotent groups.

2.1 Orientably-regular maps

A map is a cellular embedding of a connected graph or multigraph into a closed surface, giving a decomposition of the surface into 0-cells (the vertices), 1-cells (the edges), and 2-cells (called the faces of the map). Note that the faces must be simply-connected (that is, homeomorphic to a unit disc in \(\mathbb {R}^2\)), which forces connectedness of the underlying graph. A map is said to be orientable if its carrier surface is orientable.

An automorphism of a map \(\mathcal {M}\) is a permutation of the cells of \(\mathcal {M}\) that sends vertices to vertices, edges to edges, and faces to faces, and preserves incidence between them. The group of all automorphisms of \(\mathcal {M}\) is denoted by \(\mathrm{Aut}(\mathcal {M})\). In the orientable case, an automorphism may preserve or reverse orientation, but in this paper, we will deal with only orientable maps, and we will consider only orientation-preserving automorphisms. These form a group \(\mathrm{Aut}^+(\mathcal {M})\), which has index 1 or 2 in \(\mathrm{Aut}(\mathcal {M})\), depending on whether the map is chiral or reflexible, respectively.

It is easy to see that by connectedness, every orientation-preserving automorphism is uniquely determined by its effect on any given ordered edge (or arc, or dart) of \(\mathcal {M}\), and so \(\mathrm{Aut}^+(\mathcal {M})\) acts semi-regularly on the set \(D_{\mathcal {M}}\) of all darts of \(\mathcal {M}\). It follows that the largest conceivable number of orientation-preserving automorphisms is equal to \(|D_{\mathcal {M}}|\), and that this maximum is achieved if and only if the action of \(\mathrm{Aut}^+(\mathcal {M})\) is transitive (and hence regular) on \(D_{\mathcal {M}}\). Thus \(\mathcal {M}\) is said to be regular (or to be more precise, orientably-regular) if \(\mathrm{Aut}^+(\mathcal {M})\) is transitive on \(D_{\mathcal {M}}\), or equivalently, if \(|\mathrm{Aut}^+(\mathcal {M})|=|D_{\mathcal {M}}|\).

When \(\mathcal {M}\) is orientably-regular, the group \(\mathrm{Aut}^+(\mathcal {M})\) also acts transitively on vertices, and on edges, and on faces of \(\mathcal {M}\). It follows that every vertex has the same degree/valency q, and every face has the same size p, and in this case, we say that \(\mathcal {M}\) has type \(\{p,q\}\).

Next, it is well known that every orientably-regular map \(\mathcal {M}\) can be identified with a triple \((G;\, a,b)\), where G is a finite group and \(\{a,b\}\) is a generating pair for G with \(b^2=1\). The easiest way to do this is to take \(G = \mathrm{Aut}^+(\mathcal {M})\), and then let a be an automorphism of \(\mathcal {M}\) that fixes a vertex v and induces a single-step rotation of its q neighbours, consistently with the orientation of the surface, and let b be an automorphism of \(\mathcal {M}\) that interchanges v with one of its neighbours, reversing the edge e between them. In that case, the product ab preserves a face f containing e (and v), and induces a single-step rotation of the p vertices of f, again consistently with the orientation of the surface. Then a and b satisfy the defining relations of the ordinary triangle group \(\Delta (q,2,p)\), namely \(a^q = b^2 = (ab)^p = 1\), and also by connectedness, they generate G.

Any such triple \((G;\, a,b)\) may be called an algebraic orientably-regular map. From any such triple we can construct an orientably-regular map \(\mathcal {M}\), as follows. The vertices, edges, and faces of \(\mathcal {M}\) may be taken as the (right) cosets in G of the subgroups \(V = \langle a \rangle \), \(E = \langle b \rangle \) and \(F = \langle ab \rangle \), respectively, with incidence between given by non-empty intersection of cosets. In particular, the number of vertices of \(\mathcal {M}\) is equal to the index of \(V = \langle a \rangle \) in G. Furthermore, the group G acts naturally on \(\mathcal {M}\) by right multiplication of cosets, preserving incidence, and indeed \(G = \mathrm{Aut}^+(\mathcal {M})\).

If the underlying graph of this map \(\mathcal {M}\) has multiple edges, then some non-trivial subgroup of \(V = \langle a \rangle \) is preserved by b, and so the core of V in G is non-trivial. The converse also holds, and therefore the map \(M = (G;\, a,b)\) has simple underlying graph if and only if \(V = \langle a \rangle \) is core-free in G.

Also the genus of the carrier surface of \(\mathcal {M}\) is the non-negative integer g given by the Euler-Poincaré formula \(2-2g = |G|(1/q-1/2+1/p)\).

An easy example of an algebraic orientably-regular map \(\mathcal {M}=(G;\, a,b)\) occurs when G is a cyclic group of order n, generated by the element a, and b is the identity element of G. This map has a single vertex and n darts, with each dart being self-reverse.

Now let \(\mathcal {M}_1=(G_1;\, a_1,b_1)\) and \(\mathcal {M}_2=(G_2;\, a_2,b_2)\) be algebraic orientably-regular maps. If there exists a group epimorphism \(\phi : G_1\rightarrow G_2\) taking \(a_1\mapsto a_2\) and \(b_1\mapsto b_2\), then we call \(\phi \) a map homomorphism from \(\mathcal {M}_1\) to \(\mathcal {M}_2\), and say that \(\mathcal {M}_2\) is a quotient of \(\mathcal {M}_1\), and \(\mathcal {M}_1\) is a cover of \(\mathcal {M}_2\). (Note: we are not requiring the valency of vertices to be preserved.)

It is sometimes helpful to extend the symbol for the type of an orientably-regular map \(\mathcal {M}\) by adding the size of a Petrie polygon. A Petrie path is a walk in the underlying graph of \(\mathcal {M}\) that is like a ‘zig-zag’, having the property that any two but no three consecutive edges belong to a face of \(\mathcal {M}\), and a Petrie polygon is a closed Petrie path. An orientably-regular map \(\mathcal {M}=(G;\, a, b)\) of type \(\{p,q\}\) with Petrie polygons of size r is said to have type \(\{p,q\}_r\). It is easy to see that r is twice the order of the commutator \([a,b] = a^{-1}bab\).

The mirror image of an orientably-regular map \(\mathcal {M}\) is obtained by reversing its orientation. In particular, if \(\mathcal {M}=(G;\, a,b)\) then the mirror image of \(\mathcal {M}\) is isomorphic to \((G;\, a^{-1},b)\), and has the same type as \(\mathcal {M}\). Similarly, the (geometric) dual of \(\mathcal {M}\) is obtained by interchanging the roles of vertices and faces. In particular, when \(\mathcal {M}=(G;\, a,b)\) has type \(\{p,q\}_r\), the dual of \(\mathcal {M}\) is also orientably-regular, of type \(\{q,p\}_r\), and is isomorphic to \((G;\, ab,b)\). The mirror-dual of \(\mathcal {M}\) is the mirror image of the dual of \(\mathcal {M}\) (or equivalently, the dual of the mirror image of \(\mathcal {M}\)), and also has the same type as the dual of \(\mathcal {M}\).

For more information about regular maps and the associated groups, see [5]. For more specific details of some of the above properties and constructions, see [13, 17].

2.2 Nilpotent groups

We use standard concepts and notation from group theory. We let |H| denote the order of a subgroup H of a group G, and let |G : H| denote its index in G, and we let |x| denote the order of an element x of G. Also we use [x, y] as an abbreviation for the commutator \(x^{-1}y^{-1}xy\) of an ordered pair (x, y) of elements of G, and [H, K] for the subgroup generated by all commutators [x, y] with \(x \in H\) and \(y \in K,\) when H and K are subgroups of G.

We define higher-order commutators inductively by setting

$$\begin{aligned} {[}x_1,x_2,\dots ,x_{i+1}{]}=[[x_1,x_2,\dots ,x_i],x_{i+1}] \ \hbox { for } i \ge 1. \end{aligned}$$

In the next section, we will use commutator identities (such as \([a,b]^2 = [a,b^2] [b,[a,b]]\)) in the lead-up to the proofs of our main theorems.

Next, for any group G, set \(G_0 = G\) and then define \(G_{n+1} = [G,G_n]\), for all \(n \ge 0\), so that \(G = G_0 \ge G_1 \ge G_2 \ge \cdots \ \) is the lower central series of G, with each \(G_n\) normal in G. (Note that some authors denote \(G_n\) by \(\gamma _{n+1}(G)\) or \(L_{n+1}(G)\) or similar.)

It is easy to prove that if G is generated by a subset X, then \(G_n\) is generated by all conjugates in G of elements of \([x_1,x_2,\dots ,x_{n+1}]\) with \(x_i \in X\) for \(1\le i \le n+1\), or equivalently, by \(G_{n+1}\) and all such commutators; see [10, Hilfsatz III.1.11] for example.

The group G is said to be nilpotent if \(G_c\) is trivial for some c, and then the smallest c for which this happens is called the nilpotency class of G. Also (in general) for each \(n \ge 0\), we may consider the quotient \(G/G_n\). For example, \(G/G_1 = G/[G,G]\) is the abelianisation of G, and if \(G_{n-1} \ne G_n\), then \(G/G_n\) is called the maximal class n nilpotent quotient of G.

Every finite p-group is nilpotent, and every direct product of nilpotent groups is nilpotent. In fact, an equivalent definition for finite groups is that G is nilpotent if and only if G is the direct product of its Sylow subgroups; see [10, Hauptsatz III.2.3].

3 Nilpotent maps

In this section we prove our main theorems. We establish an upper bound on the number of vertices of a nilpotent regular map of class c. In doing so, we derive an exact formula for the maximum number of vertices in such a map, and show that there is exactly one nilpotent map \(\mathcal {M}_c\) of class c with simple underlying graph attaining this maximum, and derive some of its properties.

We begin with the observation that in proving the bound we may restrict our attention to maps whose automorphism group is a 2-group. The following lemma is essentially a rephrasing of Theorem 3.2 from [16].

Lemma 3.1

Let \(\mathcal {M}=(G;\, a,b)\) be an orientably-regular map for which \(G = \mathrm{Aut}^+(\mathcal {M})\) is nilpotent. Then \(\mathcal {M}\) has a quotient \(\mathcal {M}' = (H;\,a',b')\) with the property that H is a 2-group and \(\mathcal {M}'\) has the same number of vertices as \(\mathcal {M}\).

Proof

Since G is nilpotent, G is isomorphic to the direct product of its Sylow subgroups, and so can be expressed as a direct product \(H \times K\), where H is the Sylow 2-subgroup of G and K is the Hall \(2'\)-subgroup of G (of odd order).

Now the canonical generators of \(G = \mathrm{Aut}^+(\mathcal {M})\) can be written as \(a=a'a''\) and \(b=b'b''\), where \(\langle a',b'\rangle =H\) and \(\langle a'',b''\rangle =K\), but since \(b^2 = 1\) and K has odd order, we must have \(b''=1\), and therefore K is cyclic, generated by \(a''\). It follows that the subgroup H determines an orientably-regular map \(\mathcal {M}' = (H;\, a',b')\) whose automorphism is a 2-group. Also since H and K have relatively prime orders, we find that \(|G|/|\langle a\rangle | = |H||K|/(|\langle a'\rangle ||\langle a''\rangle |) = |H|/|\langle a'\rangle |\), and so the maps \(\mathcal {M}\) and \(\mathcal {M}'\) have the same number of vertices.

Finally, it is easy to see that \(\mathcal {M}'\) is isomorphic to \((G/K;\, aK, bK)\), and hence a quotient of \(\mathcal {M}\), as required. \(\square \)

Now let \(\Gamma = \mathbb {Z}* C_2\) be the free product of the infinite cyclic group with the cyclic group of order 2, with presentation \(\langle \, x,y \mid y^2 = 1 \,\rangle \).

Clearly, every finite homomorphic image G of this group \(\Gamma \) gives rise to an algebraic map \(\mathcal {M}= (G;a,b)\), where a and b are the images of x and y, respectively. When investigating a bound for the number of vertices of (G; a, b) with G nilpotent, we may restrict our attention to the case where the vertex-stabiliser \(V = \langle a \rangle \) is core-free in G. By the proof of Lemma 3.1, in that case G is a 2-group.

Before we proceed further, we introduce two infinite families of nilpotent quotients of \(\Gamma \):

Example 3.2

For each \(m \ge 2\), the dihedral group \(D_m = \langle \, a,b \ | \ a^m = b^2 = (ab)^2 = 1 \,\rangle \) is a quotient of \(\,\Gamma \) via the epimorphism taking (x, y) to (a, b). Also \(D_m\) is nilpotent whenever m is a power of 2; indeed, if \(m = 2^k\) then \(D_m\) has nilpotency class k. On the other hand, for all \(m \ge 2\) the subgroup generated by a is normal, and hence not core-free in \(D_m\).

Example 3.3

For any \(n \ge 1\), let \(m = 2^n\) and let P be the wreath product \(C_m \wr S_2\), which is generated by two commuting elements u and v of order m and an involution t that interchanges u and v by conjugation. Now \(u^t = v\), so \(\langle u,t \rangle = P\), and u and t have orders m and 2, so P is a quotient of \(\Gamma \). Next, \([u,t] = u^{{-1}}\, u^t = u^{{-1}}v = w\), say, and this has order m, and is centralised by u and inverted by (conjugation by) t, so w generates a cyclic normal subgroup of P. It follows that \(P_1 = [P,P]\) is generated by w, and an easy induction shows that \(P_j\) is the subgroup generated by \(w^{2^{j{-1}}}\), for all j. The smallest j for which \(P_j\) is trivial is \(n+1\), since w has order \(m = 2^n\), and so P has nilpotency class \(n+1\). Also t conjugates each non-trivial power of u to the same non-trivial power of v, and in particular, t does not centralise the involution \(u^{m/2} = u^{2^{n{-1}}}\). Hence the core K of the cyclic subgroup generated by u is trivial.

Note The regular map associated with Example 3.2 is the dual of the standard embedding of the m-cycle on the sphere. In Example 3.3 the underlying graph of the associated regular map is the complete bipartite graph \(K_{m,m}\) and the map is known as the standard embedding of \(K_{m,m}\). The associated surface is the Fermat curve \(x^m+y^m=1\); see [14].

We now proceed with some theory of nilpotent quotients of the group \(\Gamma = \mathbb {Z}* C_2\).

Lemma 3.4

The factor group \(\Gamma _n/\Gamma _{n+1}\) is non-trivial and abelian, for all \(n \ge 0\).

Proof

First, the derived subgroup of \(\Gamma _n\) is contained in \([\Gamma ,\Gamma _n] = \Gamma _{n+1}\), and so \(\Gamma _n/\Gamma _{n+1}\) is abelian, for all \(n \ge 0\). To see that \(\Gamma _n/\Gamma _{n+1}\) is non-trivial, we note that the dihedral group \(D_{2^{n+1}} = \langle \, a,b \ | \ a^{2^{n+1}} = b^2 = (ab)^2 = 1 \,\rangle \) is a quotient of \(\Gamma \), and is nilpotent of class \(n+1\). It follows that \(D_{2^{n+1}}\) is a quotient of \(\Gamma /\Gamma _{n+1}\) but not of \(\Gamma /\Gamma _n\), so \(\Gamma _n \ne \Gamma _{n+1}\), and \(\Gamma _n/\Gamma _{n+1}\) is non-trivial.

Even more can be said about the factors \(\Gamma _n/\Gamma _{n+1}\) when \(n \ge 1\).

Proposition 3.5

For each \(n \ge 1\), the factor \(\Gamma _n/\Gamma _{n+1}\) of the lower central series of \(\Gamma \) is a finite elementary abelian 2-group.

Proof

First, we know that \(\Gamma _n/\Gamma _{n+1}\) is abelian (by Lemma 3.4), and also that \(\Gamma _n/\Gamma _{n+1}\) is finitely generated, since \(\Gamma _n\) is generated by \(\Gamma _{n+1}\) and commutators \([x_1,x_2,\dots ,x_{n+1}]\) with \(x_i \in X = \{x,y\}\) for \(1\le i \le n+1\) (by [10, Hilfsatz III.1.11]). Hence all we have to do is to show that \(\Gamma _n/\Gamma _{n+1}\) has exponent 2. On the other hand, \(\Gamma _n\) is generated by conjugates of elements of the form [a, b], where a and b are elements of \(\Gamma \) and \(\Gamma _{n{-1}}\), respectively, and so this reduces to showing that the square of each such [a, b] lies in \(\Gamma _{n+1}\). To do this, we use the easily proved fact that \(\, [a,b]^2 = [a,b^2] [b,[a,b]]\,\) for all a and b (in a given group).

Now \(\Gamma _1\) is generated by all conjugates of \(z = [x,y]\), since adjoining the relation \([x,y] = 1\) to the presentation for the 2-generator group \(\Gamma \) gives the abelianisation \(\Gamma /[\Gamma ,\Gamma ] = \Gamma /\Gamma _1\). Also if \(w \in \Gamma \) then \(w^{{-1}}zw = z[z,w] = z[[x,y],w] \in z[\Gamma _1,\Gamma ] = z\Gamma _2\), and therefore every conjugate of z lies in the coset \(z\Gamma _2\). Thus \(\Gamma _1/\Gamma _2\) is cyclic, generated by \(z\Gamma _2\). Moreover, if we take \(a = x\) and \(b = y\) in the commutator identity \([a,b]^2 = [a,b^2] [b,[a,b]]\), we find that \(z^2 = [x,y]^2 = [x,y^2] [y,[x,y]] = [y,[x,y]] \in [\Gamma ,\Gamma _1] = \Gamma _2\). Thus \(\Gamma _1/\Gamma _2\) has order 2.

Next, suppose \(n > 1\), and let a and b be elements of \(\Gamma \) and \(\Gamma _{n{-1}}\), respectively. Then by induction, we may assume that \(\Gamma _{n{-1}}/\Gamma _n\) has exponent 2, and hence that \(b^2 \in \Gamma _n\). It follows that \([a,b^2] \in [\Gamma ,\Gamma _n] = \Gamma _{n+1}\), and then since also \([b,[a,b]] \in [\Gamma ,[\Gamma ,\Gamma _{n{-1}}]] = [\Gamma ,\Gamma _n] = \Gamma _{n+1}\), the above commutator identity gives \([a,b]^2\) as the product of a pair of elements of \(\Gamma _{n+1}\), so \([a,b]^2 \in \Gamma _{n+1}\). Hence the square of every generator of \(\Gamma _n\) lies in \(\Gamma _{n+1}\), and so \(\Gamma _n/\Gamma _{n+1}\) has exponent 2, as required.\(\square \)

Note that Proposition 3.5 does not hold for \(n = 0\), because \(\Gamma _0/\Gamma _1 = \Gamma /[\Gamma ,\Gamma ]\) is the abelianisation of \(\Gamma \), which is isomorphic to \({\mathbb Z} \oplus C_2\) (since x has infinite order).

Corollary 3.6

\((xy)^{2^n} \in x^{2^n}\Gamma _n\) for all \(n \ge 1\).

Proof

We prove this by induction on n. First \((xy)^2 = x^{2}x^{{-1}}yxy = x^{2}[x,y] \in x^{2}\Gamma _1\). Next, suppose \((xy)^{2^n} = x^{2^{n}}z\) where \(z \in \Gamma _n\). Then

$$\begin{aligned} (xy)^{2^{n+1}} = (x^{2^{n}}z)^2 = (x^{2^{n}})^2 x^{{-2}^{n}}z x^{2^{n}}z^{{-1}}z^2 = x^{2^{n+1}} [x^{2^{n}},z^{{-1}}] z^2, \end{aligned}$$

and as \([x^{2^{n}},z^{{-1}}] \in [\Gamma ,\Gamma _n] = \Gamma _{n+1}\) and also \(z^2 \in \Gamma _{n+1}\) (because \(\Gamma _j/\Gamma _{j+1}\) has exponent 2), it follows that \((xy)^{2^{n+1}} \in x^{2^{n+1}}\Gamma _{n+1}\). \(\square \)

The first factor \(\Gamma /\Gamma _1\) of the lower central series of \(\Gamma \) is the abelianisation of \(\Gamma \), which is isomorphic to \({\mathbb Z} \oplus C_2\) since x has infinite order while y has order 2. The second factor \(\Gamma _1/\Gamma _2\) is cyclic of order 2, as shown in the proof of Proposition 3.5.

The ranks of the factor groups \(\Gamma _n/\Gamma _{n+1}\) for \(n > 1\) are given by some work by Labute [15] on the associated Lie algebra. These ranks are determined by a sequence of integers and two related sequences of partial sums, defined as follows:

Set \(h_1=0\), and then for \(n\ge 2\) take

$$\begin{aligned} h_n&= \frac{1}{n}\sum _{d|n} \mu (n/d)\left( \sum _{0\le i\le d/3}({-1})^i\frac{d}{d{-2}i}\left( {\begin{array}{c}d{-2}i\\ i\end{array}}\right) 2^{d-3i}\right) ,\\ g_n&= h_1 + h_2 + \dots + h_n=\sum _{j=1}^n h_n,\\ f_n&= g_1 + g_2 + \dots + g_n=\sum _{i=1}^n g_n, \end{aligned}$$

where \(\mu \) denotes the Möbius function. The first few terms of these three sequences are given below, each starting from \(n = 1\):

\(h_n\) :

:   0, 1, 1, 1, 2, 2, 4, 5, 8, 11, 18, 25, 40, 58, 90, 135, 210, 316, ...

\(g_n\) :

:   0, 1, 2, 3, 5, 7, 11, 16, 24, 35, 53, 78, 118, 176, 266, 401, 611, 927, ...

\(f_n\) :

:   0, 1, 3, 6, 11, 18, 29, 45, 69, 104, 157, 235, 353, 529, 795, 1196, 1807, 2734, ....

We note that the sequence \(h_n\) above agrees in all but the first term with the sequence A006206 in Neil Sloane’s On-line Encyclopedia of Integer Sequences [20], which counts (among other things) the number of aperiodic binary necklaces of length n with no subsequence 00, excluding the single-term necklace ‘0’.

The main theorem we use from the work of Labute is the following:

Theorem 3.7

(Labute [15]) The rank of the factor group \(\Gamma _{n{-1}}/\Gamma _{n}\) is \(g_n\), for all \(n \ge 2\).

An immediate consequence is the following:

Corollary 3.8

The order of the quotient \(\Gamma _1/\Gamma _n\) is \(2^{f_{n}}\), for all \(n\ge 2\).

Proof

\(|\Gamma _1 : \Gamma _n| = |\Gamma _1 : \Gamma _2| |\Gamma _2 : \Gamma _3| \dots |\Gamma _{n{-1}} : \Gamma _n| = 2^{g_2+g_3+ \,\cdots \, +g_n} = 2^{f_{n}-f_{1}} = 2^{f_{n}}.\) \(\square \)

The above information will help us find the orders of the largest nilpotent simple maps of given class.

Before continuing further, we define \(H^{(n)}\) to be the class n nilpotent quotient \(\Gamma /\Gamma _n\) of \(\Gamma \). Also we need the following lemma, concerning the generators x and y of \(\Gamma \):

Lemma 3.9

\( [x^{2k},y] = [x^k,[y,x^k]] [[y,x^k],y]\) for all k.

Proof

Since \(y^2 = 1\), we have

$$\begin{aligned}{}[x^k,[y,x^k]] [[y,x^k],y]= & {} x^{-k} [x^k,y] x^{k} [y,x^k] [x^k,y] y [y,x^k] y\\= & {} x^{-k} [x^k,y] x^{k} y [y,x^k] y \\= & {} x^{-k} x^{-k}yx^{k}y x^{k} y yx^{-k}yx^{k} y = x^{{-2}k}yx^{2k}y = [x^{2k},y]. \end{aligned}$$

\(\square \)

Proposition 3.10

For \(n \ge 0\), the subgroup of \(H^{(n+1)} = \Gamma /\Gamma _{n+1}\) generated by the image of \(\,x^{2^n}\) is normal, but for \(n > 0\), the subgroup generated by the image of \(\,x^{2^{n{-1}}}\) is not.

Proof

First, we show that \([x^{2^i},y]\) lies in \(\Gamma _{i+1}\), for all \(i \ge 0\). This is obvious for \(i = 0\), and we can prove it easily by induction for all \(i > 0\): if \(k = 2^{i{-1}}\) and \([x^k,y] \in \Gamma _i\), then \([x^k,[y,x^k]] \in [\Gamma ,\Gamma _i] = \Gamma _{i+1}\) and \([[y,x^k],y] \in [\Gamma _i,\Gamma ] = \Gamma _{i+1}\), and so Lemma 3.9 implies that \([x^{2^i},y] = [x^{2k},y]\) is a product of a pair of elements of \(\Gamma _{i+1}\).

It follows that \(y^{{-1}}(x^{2^n})y = x^{2^n}[x^{2^n},y]\) lies in the coset \(x^{2^n}\Gamma _{n+1}\), for all n. Hence in the quotient \(\Gamma /\Gamma _{n+1}\), the image of y normalises the cyclic subgroup \(X_n/\Gamma _{n+1}\) generated by \(x^{2^n}\Gamma _{n+1}\). Since \(x^{2^n}\) is centralised by x, the latter subgroup is also centralised by the image of x, and so we find that \(X_n/\Gamma _{n+1} \lhd \Gamma /\Gamma _{n+1}\).

On the other hand, suppose that the subgroup generated by the image of \(\,x^{2^{n{-1}}}\) in \(\Gamma /\Gamma _{n+1}\) were normal. Then in the quotient \(C_{2^n} \wr S_2\) of \(\Gamma \) considered in Example 3.3, the subgroup generated by \(u^{2^{n{-1}}}\) would be normal, but it is not—contradiction. \(\square \)

For a quotient of \(\Gamma \) to be the group of a regular map with simple underlying graph, we need the stabiliser of a vertex to be core-free. It follows that the largest nilpotent quotient of class c that is admissible is the quotient \(U^{(c)} = H^{(c)}/K\), where K is the core of the subgroup generated by the image of x in \(H^{(c)} = \Gamma /\Gamma _c\), and every other admissible quotient that is nilpotent of class at most c with a core-free vertex-stabiliser is a quotient of \(U^{(c)}\).

Theorem 3.11

For each \(c \ge 1\), the group \(U^{(c)} = H^{(c)}/K\) has order \(2^{c+f_{c}}\), where \(f_c\) is as given just before Theorem 3.7 . The corresponding regular map has type \(\{2^c,2^{c{-1}}\}_{2^c}\), with valency \(2^{c{-1}}\), face-size \(2^c\), and Petrie polygons of length \(2^c\).

Proof

By Proposition 3.10 we know that the core K of the cyclic subgroup \(X_c/\Gamma _c\) generated by the image of x in \(H^{(c)} = \Gamma /\Gamma _c\) is generated by the image of \(\,x^{2^{c{-1}}}\). It follows that in the quotient \(U^{(c)} = H^{(c)}/K\), the order of the image of x is \(2^{c{-1}}\). Also because the subgroup generated by x has trivial intersection with both \(\Gamma _1\) and the subgroup generated by y, the abelianisation of \(U^{(c)}\) has order \(2^{c{-1}} \cdot 2 = 2^c\). Next, by Corollary 3.8 the factor group \(\Gamma _1/\Gamma _c\) has order \(2^{f_c}\), and hence so does its image in the quotient \((\Gamma /\Gamma _c)/K = U^{(c)},\) which is the derived subgroup \((U^{(c)})_{1}\). Thus \(|\,U^{(c)}| = |\,U^{(c)}:(U^{(c)})_{1}| |(U^{(c)})_{1}| = 2^{c} \cdot 2^{f_c} = 2^{c+f_c}\).

We now calculate the orders of the images of xy and [x, y] in \(U^{(c)},\) which determine the sizes of faces and Petrie polygons in the resulting map. By Corollary 3.6, we know that \((xy)^{2^c}\) lies in \(\Gamma _c\), and so the order of the image of xy in \(U^{(c)}\) divides \(2^c\). Similarly, the commutator [x, y] lies in \(\Gamma _1\), and the exponent of \(\Gamma _1/\Gamma _c\) divides \(2^{c{-1}}\) (since \(\Gamma _j/\Gamma _{j+1}\) has exponent 2 for \(1 \le j < c\)), and so the order of the image of [x, y] in \(U^{(c)}\) divides \(2^{c{-1}}\). On the other hand, consider the class c nilpotent quotient \(C_{2^{c{-1}}} \wr S_2\) of \(\Gamma \) in Example 3.3 (with \(n = c{-1}\)). In this group, the images of xy and [x, y] are the elements ut and [u, t], and these have orders \(2m = 2^c\) and \(m = 2^{c{-1}}\), respectively (since \((ut)^2 = uu^t = uv\) and \([u,t] = u^{{-1}}u^t = u^{{-1}}v\)). It follows that the orders of the images of xy and [x, y] in \(U^{(c)}\) are exactly \(2^c\) and \(2^{c{-1}}\), and hence in the corresponding map, the face-size is \(2^c\), and the length of Petrie polygons is \(2 \cdot 2^{c{-1}} = 2^c\) as well.   \(\square \)

Corollary 3.12

For every integer \(c\ge 1\) there exists a unique nilpotent regular map \(\mathcal {M}_c\) of class c with simple underlying graph, having \(2^{1+f_c}\) vertices, and type \(\{2^c,2^{c{-1}}\}_{2^c}\). Furthermore, this map is ‘universal’, in the sense that every nilpotent regular map of class at most c with simple underlying graph is a quotient of \(\mathcal {M}_c\).

We note that by its universality, the map \({\mathcal M}_c\) is reflexible, and also self-Petrie (isomorphic to its Petrie dual—see [5] or [21]). In fact, universality of \({\mathcal M}_c\) implies also that for every odd j the map \({\mathcal M}_c\) is invariant under the Coxeter/Wilson ‘hole’ operator \(H_j\), which replaces the image \(\bar{x}\) of the generator x by its j-th power \({\bar{x}}^j\) (see [21]). Similarly, \({\mathcal M}_c\) is invariant under every ‘e-switch’, which replaces the local rotations at all vertices in one part of a bipartite map by their e-th powers (see [18]). The cases \(j = -1\) and \(e = -1\) correspond to reflection and the Petrie dual operator, respectively.

4 Nilpotent simple regular maps of small class

We list below all of the nilpotent maps of class 2 to 4, having simple underlying graph. There are 26 such maps, and for each one, we give its genus g, its type \(\{p,q\}_r\) and the order of its orientation-preserving automorphism group \(G = \mathrm{Aut}^+(\mathcal {M})\), as well as defining relators for G (as words in the given generators x and y of \(\Gamma = \mathbb {Z}* C_2 = \langle \, x,y \ | \ y^2 = 1 \,\rangle \)). These were found with the help of the Magma system [2]. The last one for each class c is the universal map \(\mathcal {M}_c\). All of these maps are reflexible.

c

g

\(\{p,q\}_r\)

|G|

Defining relators for G

2

0

\(\{4,2\}_4\)

8

\( x^{2}, y^{2}, (xy)^{4} \)

3

0

\(\{8,2\}_8\)

16

\( x^{2}, y^{2}, (xy)^{8} \)

3

1

\(\{4,4\}_4\)

32

\( x^{4}, y^{2}, (xy)^{4}, [x,y]^{2} \)

3

3

\(\{8,4\}_8\)

32

\( x^{4}, y^{2}, (xy)^{2}(x^{-1}y)^{2} \)

3

5

\(\{8,4\}_8\)

64

\( x^{4}, y^{2}, xyx^{2}yx^{-1}yx^{-2}y, (xy)^{4}(x^{-1}y)^{4} \)

4

0

\(\{16,2\}_{16}\)

32

\( x^{2}, y^{2}, (xy)^{16} \)

4

1

\(\{4,4\}_8\)

64

\( x^{4}, y^{2}, (xy)^{4}, (x^{2}y)^{2}(x^{-2}y)^{2} \)

4

5

\(\{8,4\}_4\)

64

\( x^{4}, y^{2}, [x,y]^{2}, (x^{2}y)^{2}(x^{-2}y)^{2} \)

4

7

\(\{16,4\}_{16}\)

64

\( x^{4}, y^{2}, xyx^{2}yx^{-1}yx^{-2}y, (xy)^{6}(x^{-1}y)^{2} \)

4

9

\(\{8,4\}_8\)

128

\( x^{4}, y^{2}, (x^{2}y)^{2}(x^{-2}y)^{2}, xyx^{2}yxyx^{-1}yx^{-2}yx^{-1}y, (xy)^{4}(x^{-1}y)^{4} \)

4

13

\(\{16,4\}_{16}\)

128

\( x^4, y^2, xyx^{2}yx^{-1}yx^{-2}y, (xy)^{8}(x^{-1}y)^{8} \)

4

13

\(\{16,4\}_{16}\)

128

\( x^4, y^2, (x^{2}y)^{2}(x^{-2}y)^{2}, xyx^{2}yxyx^{-1}yx^{-2}yx^{-1}y, (xy)^{5}x^{-1}yxyx^{-1}y \)

4

21

\(\{16,8\}_{16}\)

128

\( x^8, y^2, (xy)^{2}(x^{-1}y)^{2} \)

4

21

\(\{16,8\}_{16}\)

128

\( x^8, y^2, xyx^{2}yx^{-1}yx^{-2}y, x^{3}yx^{-1}yxyx^{-3}y \)

4

25

\(\{16,4\}_{16}\)

256

\( x^4, y^2, (x^{2}y)^{2}(x^{-2}y)^{2}, xyx^{2}yxyx^{-1}yx^{-2}yx^{-1}y,\)

    

   \((xy)^{4}(x^{-1}y)^{2}(xy)^{2}(x^{-1}y)^{2}(xy)^{2}(x^{-1}y)^{4} \)

4

33

\(\{8,8\}_{8}\)

256

\( x^8, y^2, xyx^{2}yx^{-1}yx^{-2}y, (xy)^8 \)

4

33

\(\{8,8\}_{8}\)

256

\( x^8, y^2, x^{3}yx^{-2}yxyx^{-2}y, (xy)^8 \)

4

41

\(\{16,8\}_{16}\)

256

\( x^8, y^2, xyx^{2}yx^{-1}yx^{-2}y, (xy)^{4}(x^{-1}y)^{4} \)

4

41

\(\{16,8\}_{16}\)

256

\( x^8, y^2, x^{3}yx^{-2}yxyx^{-2}y, (xy)^{4}(x^{-1}y)^{4} \)

4

65

\(\{8,8\}_{8}\)

512

\( x^8, y^2, (x^{2}y)^{2}(x^{-2}y)^{2}, xyx^{4}yx^{-1}yx^{-4}y, (xyxyx^{-2}y)^2, (xy)^8 \)

4

81

\(\{16,8\}_{16}\)

512

\( x^8, y^2, xyx^{2}yx^{-1}yx^{-2}y, (xy)^{5}(x^{-1}y)^{4}(xy)^{3}(x^{-1}y)^{4} \)

4

81

\(\{16,8\}_{16}\)

512

\( x^8, y^2, (x^{2}y)^{2}(x^{-2}y)^{2}, xyx^{4}yx^{-1}yx^{-4}y, (xyxyx^{-2}y)^2, (xy)^{4}(x^{-1}y)^{4} \)

4

81

\(\{16,8\}_{16}\)

512

\( x^8, y^2, x^{3}yx^{-2}yxyx^{-2}y, \)

    

   \((xy)^{3}(x^{-1}y)^{2}xy(x^{-1}y)^{3}(xy)^{2}x^{-1}y(xy)^{2}(x^{-1}y)^{2} \)

4

81

\(\{16,8\}_{16}\)

512

\( x^8, y^2, (x^{2}y)^{2}(x^{-2}y)^{2}, xyx^{4}yx^{-1}yx^{-4}y, (xyxyx^{-2}y)^2,\)

    

   \((xy)^{3}(x^{-1}y)^{2}xy(x^{-1}y)^{2} \)

4

81

\(\{16,8\}_{16}\)

512

\( x^8, y^2, (x^{2}y)^{2}(x^{-2}y)^{2}, xyx^{4}yx^{-1}yx^{-4}y, (xyxyx^{-2}y)^2, \)

    

   \(x^{3}yx^{-1}yxyx^{-1}yxy(x^{-1}y)^{3} \)

4

161

\(\{16,8\}_{16}\)

1024

\( x^8, y^2, (x^{2}y)^{2}(x^{-2}y)^{2}, xyx^{4}yx^{-1}yx^{-4}y, (xyxyx^{-2}y)^2, \)

    

   \((xy)^{5}(x^{-1}y)^{2}xy(x^{-1}y)^{3}(xy)^{2}(x^{-1}y)^{3} \)

For class 5, there are 681 simple maps, of which 663 are reflexible (with genera ranging from 0 to 13313), while the remaining 18 such maps fall into nine chiral pairs (of genera 45, 105, 177, 417 (two pairs) and 833 (four pairs)). The smallest chiral examples are a pair of genus 25 and type \(\{16,4\}_8\), with \(G = \mathrm{Aut}^+(\mathcal {M})\) having order 256 and nilpotency class 6; these are the dual and mirror-dual of the chiral map labelled C25.1 in the first author’s list of all arc-transitive chiral maps of genus 2 to 101 (see [4]).