This chapter is about the reversible elements in the group G of invertible biholomorphic germs in one variable and some of its subgroups. The theory of reversibility for formally-reversible formal power series in one variable has already been dealt with in Chapter 10. In the present chapter, we shall denote the group of these formal series by G. The group G is a subgroup of G, which implies that reversible biholomorphic germs are formally reversible.

A priori, biholomorphic conjugacy is a much finer relation than formal conjugacy, so one expects that the formal conjugacy class of an element of G will split into many distinct biholomorphic conjugacy classes. This is often the case, and indeed we shall see (in Section 12.6) that there exist germs that are formally reversible, but not biholomorphically reversible.

Let us discuss the groups to be studied in more detail. A germ at 0 is an equivalence class of functions under the relation that regards two functions as equivalent if they agree on some neighbourhood of 0. Let S denote the set of those invertible complex holomorphic maps defined on a neighbourhood of 0 that fix 0. The group G of biholomorphic germs consists of the equivalence classes of S under the equivalence relation just described. Thus an element of G is represented by some function f, holomorphic on some neighbourhood of 0 (which depends on f) with f (0) = 0 and f′(0) ≠ 0. Two such functions represent the same germ if they agree on some neighbourhood of 0. The group operation is composition and the identity is the germ of the identity function z ↦ z, which we denote by 1.

Next we introduce notation for some important subgroups of G and for some quantities that were used in Chapter 10 to distinguish the conjugacy classes of formal power series. Each element of G can be represented by a convergent complex power series with no constant term and with a nonzero z coefficient.

Reversibility is the study of those elements of a group that are conjugate to their own inverses. Reversible maps appear naturally in classical dynamics; for instance, in the pendulum, the n-body problem, and billiards. They also arise in less obvious ways in connection with problems in geometry, complex analysis, approximation, and functional equations. When a problem has a connection to a reversible map, this opens it to attack using dynamical ideas, such as ergodic theory and the theory of flows. Reversibility has its origins in work of Birkhoff, Arnol'd, Voronin, Sevryuk, Siegel, Moser, Smale, and Devaney, among others, mainly in the context of continuous dynamical systems [8, 9, 10, 11, 31, 32, 66, 131, 211, 214, 237]. Devaney initiated the formal study of smooth reversible systems, not necessarily derived from a Hamiltonian, and there has been considerable work on such systems. The main focus has been on higher dimensions, and the systematic study of discrete reversible systems in low dimensions is more recent. We concentrate here on the discrete system theory, and on developments since the turn of the century.

The subject relates to involutions, conjugacy problems, and automorphism groups. The reversible elements of a group are those elements that are conjugate to their own inverses, and the strongly-reversible elements are those elements that are conjugate to their own inverses by involutions. Both types of element have been studied in many contexts. For finite groups, the terms real and strongly real are used instead of reversible and strongly reversible [117, Section 9.1] because of the connections with real characters. Questions of reversibility for classical groups have been addressed in works such as [82, 150, 151, 153]. The authors of these papers use the term bireflectional to describe a group comprised entirely of strongly-reversible maps. There is a rich modern literature on reversibility in dynamical systems, which includes [16, 66, 159, 160, 162, 163, 164, 165, 198, 199, 205, 211].

Reversibility

The natural context for the study of reversibility is the theory of groups. In this chapter we review the basics of reversibility, borrowing heavily from work of Aschbacher, Baake, Goodson, Lamb, and Roberts [12, 19, 110, 159, 164]. We assume throughout that G is a group with identity 1.

2.1.1 Reversible elements

An element g of a group G is said to be reversible in G if there is another element h of G such that

hgh−1 = g−1.

In finite groups, the reversible elements are called real elements because of their relationship with real characters (see Chapter 3, page 49). We denote by R(G), or just R if the context is clear, the set of reversible elements in G. The reversibility problem in G is the problem of characterising the reversible elements of G.

If the equation hgh−1 = g−1 is satisfied, then we say that h reverses g and h is a reverser for g. We denote by Rg(G), or just Rg, the set of reversers for g, which may of course be empty. We note that h reverses g if and only if ghg = h; that is, the reversers of g are the fixed points of the bijection x ↦ gxg.

We denote the set of involutions in G (elements of order at most two) by I(G), or just I. Thus

I = {g ∈ G : g2 = 1}.

Example 2.1

1. (i) The identity 1 is reversible, and every element of G reverses 1.

2. (ii) An element of G is an involution if and only if it is reversed by 1.

3. (iii) If G is an abelian group, then R = I and R is a subgroup of G. As a special case, if G is a cyclic group of even order, then it has two reversible elements: the identity and the element of order two.

4. […]

Isometries of spherical, Euclidean, and hyperbolic space

For each positive integer n, there are three simply-connected, complete Rie-mannian n-manifolds with constant curvature, namely n-dimensional spherical space Sn, n-dimensional Euclidean space ℝn, and n-dimensional hyperbolic space ℍn. Note that ℝ1 and ℍ1 are isometric, but otherwise there are no repetitions in this list. We denote the isometry groups of these manifolds by Isom(Sn), Isom(ℝn), and Isom(ℍn). These groups are each generated by reflections. We denote the three subgroups of these three isometry groups, comprised of orientation-preserving isometries, by Isom+ (Sn), Isom+ (ℝRn), and Isom+ (ℍn). A map in Isom(Sn) lies in Isom+ (Sn) if and only if it can be expressed as a composite of an even number of reflections. Similar comments apply to the groups Isom+ (ℝn) and Isom+ (ℍn).

We studied the orthogonal group O(n, ℝ) and the special orthogonal group SO(n, ℝ) in Chapter 4; these two groups are Isom(Sn−1) and Isom+ (Sn−1), respectively. In this chapter we consider reversibility in the remaining four isometry groups Isom(ℝn), Isom(ℍn), Isom+ (ℝn), and Isom+ (ℍn).

Hyperbolic geometry in two and three dimensions

In Chapter 1 we briefly discussed reversibility in the Euclidean isometry groups Isom+ (ℝ2) and Isom+ (ℝ3). We found that, in two dimensions, the only elements that are strongly reversible, other than involutions, are translations. In three dimensions we found that all isometries are strongly reversible. Before we tackle higher-dimensional isometry groups we first, in this section, consider isometry groups of two- and three-dimensional hyperbolic space.

The classical groups

Hermann Weyl [243] coined the phrase ‘classical groups’ to describe certain subgroups of general linear groups, and other groups derived from these subgroups, that preserve particular sesquilinear forms. We discuss the collection of all classical groups in this section, but only study reversibility in a selection of them.

Let V denote a finite-dimensional vector space over a commutative field F. The general linear group of V is the group of all invertible linear transformations of V. It is denoted GL(V). Let σ denote a field automorphism of F that is an involution. A sesquilinear form on V, relative to σ, is a map B from V × V to F such that for all vectors u, v, and w in V, and scalars λ and μ in F,

(i) B(λu + μv, w) = λB(u, w) + μB(v, w)

(ii) B(w, λu + μv) = σ(λ)B(w, u) + σ(μ)B(w, v).

If σ is the identity map, then B is said to be bilinear. The sesquilinear form B is nondegenerate if each of the conditions B(u, v) = 0 for all vectors v in V, or B(v, u) = 0 for all vectors v in V, imply that u = 0. A nondegenerate sesquilinear form B is reflexive if the equation B(u, v) = 0 implies that B(v, u) = 0.

A reflexive, nondegenerate sesquilinear form V is said to be

1. ◦ alternating, if B is bilinear, and B(v, v) = 0 for each vector v in V

2. ◦ Hermitian, if σ is of order 2, and is such that B(u, v) = σ(B(v, u)) for each pair of vectors u and v

3. ◦ symmetric, if B is bilinear, and B(u, v) = B(v, u) for each pair of vectors u and v.

A famous theorem of Birkhoff and von Neumann [225, Theorem 7.1] says that each reflexive, nondegenerate sesquilinear form is either alternating, Hermitian, or symmetric. We define the isometry group of B, Isom(B), to be the subgroup of GL(V) consisting of those maps that preserve B. That is

Isom(B) = {g ∈ GL(V) : B(g(u), g(v)) = B(u, v) for each pair u and v in V}.

In this chapter, we denote the cardinality of a set F by |F|. Thus the order of a group G is denoted |G|. We recall that R(G), or just R, denotes the set of reversible elements of G, and Rg(G), or just Rg, denotes the set of elements h that satisfy hgh−1 = g−1. Also I(G), or I, denotes the set of involutions in G, and In = {τ1 … τn : τi ∈ I}.

Reversers of finite order

Proposition 3.1Suppose that elements g and h of a group G satisfy hgh−1 = g−1. If h has finite order, then either g is an involution or else the order of h is even. In the latter case, g is also be reversed by an element whose order is a power of 2.

Proof To prove the first assertion, suppose that the order of h is odd. Since each odd power of h reverses g, it follows that the identity 1 reverses g. Therefore g = g−1, so g is an involution.

Suppose now that h has even order 2kq for positive integers k and q, where q is odd. Then hq also reverses g, and this element has order 2k. This proves the second assertion.

Corollary 3.2A finite group G contains a nontrivial reversible element if and only if G has even order. In fact, if G contains a nontrivial reversible element, then it contains a nontrivial involution.

Proof Suppose first that G has a nontrivial element g that is reversed by another element h. By Proposition 3.1, one of g or h has even order, so G also has even order.

Suppose now that G has even order. By partitioning G into subsets {g, g−1} we see that G contains a nontrivial involution. Since involutions are reversible, all parts of the corollary have been accounted for.