Journal of Lie Theory Volume 9 (1999) 351–353 C ◯1999 Heldermann Verlag

Finite groups of rotations A supplement to the preceding article

Herbert Abels

Communicated by E. B. Vinberg

Abstract. This paper completes the work started in the preceding paper where the following question was asked. Given a finite set S of isometries of some affine Euclidean space. When is the group Γ generated by S discrete? In that paper we described an algorithm which reduced this question to the special case discussed here.

Suppose we are given a finite set S of rotations of Euclidean d–space. We ask when the group Γ generated by S is finite. We show here that there is an algorithm to decide this question and this algorithm has a number of steps bounded by a constant depending only on d and #S.

The main ingredient of the proof is the existence of a Zassenhaus neighbourhood Ω of the identity in a Lie group G. Recall that by definition, a neighbourhood Ω of e in G is called a Zassenhaus neighbourhood if for every discrete subgroup Γ of G the intersection Ω ∩ Γ is contained in a connected nilpotent subgroup of G, hence in an abelian compact connected subgroup if G is compact. So our approach is still another variation on a long standing and celebrated theme, namely the theorem of Jordan, which states that every finite subgroup of O_{d} contains an abelian subgroup of an index bounded by a number depending only on d. For recent work on explicit bounds and explicit Zassenhaus neighbourhoods of e in O_{d} see [2,3] and the references therein.

1. The algorithm

Theorem . There is a function f(d, m) with the following property. For every subset S of cardinality m of the group O(d) there is an algorithm with at most f(d, m) steps to decide if the subgroup of O(d) generated by S is finite. Recall that in a Lie group G a neighbourhood Ω of e is called a Zassenhaus neighbourhood of e if for every discrete subgroup Γ of G the set Γ∩Ω is contained

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag in a nilpotent connected Lie subgroup of G. Thus, if G is compact, the set Γ∩Ω is contained in a torus for every finite subgroup Γ of G. We make use of the result that every Lie group contains a Zassenhaus neighbourhood of e, cf. [4, 8.16].

We work in a fixed dimension d. Consequently, we suppress the dependence on d from the notation. Put K = O(d). Let S be a finite subset of K. We may assume that S is symmetric, i.e. S = S^{−1} , and contains the identity element e. We define inductively Γ_{1} = S, Γ_{n+1} = S ·Γ_{n} = {βγ; β ∈ S, γ ∈ Γ_{n}}.

Put Γ^{•}_{n} = Γ_{n} ∖ Γ_{n−1} .

Lemma 1. There is a constant f_{1} with the following property Γ_{n+1} ⊂ Γ_{n} · Ω for some n ≤ f_{1} .

The strange notation f_{1} for a constant should remind us that f_{1} can be considered as a function of d.

Proof. Let U be an open symmetric neighbourhood of e in K such that U^{2} ⊂ Ω. If γU ∩ Γ_{n}U = Ø for at least one γ ∈ Γ^{•}_{n+1} then vol (Γ_{n+1}U) ≥ vol (Γ_{n}U) + vol (U), where vol is a Haar measure on K. Thus f_{1} = vol (K)/vol (U) will do.

Now let n be as in Lemma 1. For every γ ∈ Γ^{•}_{n+1} choose an element β_{γ} ∈ Γ_{n} such that

α(γ) := β_{γ}^{−1} · γ ∈ Ω.

Let A be the subgroup of K generated by {α(γ) ; γ ∈ Γ^{•}_{n+1}}.

Lemma 2. Γ = Γ_{n} · A.

Proof. We have S Γ_{n} = Γ_{n+1} = Γ_{n} ∪ Γ^{•}_{n+1} ⊂ Γ_{n} ∪ Γ_{n} · A = Γ_{n}A, hence S Γ_{n}A ⊂ Γ_{n}A and thus ΓΓ_{n}A ⊂ Γ_{n}A which implies Γ ⊂ Γ_{n}A. The converse inclusion is trivial.

Corollary . Γ is finite iff the following two conditions hold.

a) The α(γ), γ ∈ Γ^{•}_{n+1} , commute.

b) Every α(γ), γ ∈ Γ^{•}_{n+1} , is of finite order.

Proof. If Γ is finite, then a) holds by definition of a Zassenhaus neighbourhood and b) is obvious. Conversely, if a) and b) hold then A is finite and hence Γ is finite by lemma 2.

This implies the theorem: Compute inductively Γ_{n}, n = 1, 2, … check if for every γ ∈ Γ^{•}_{n+1} at least one β^{−1}γ, β ∈ Γ_{n}, is contained in Ω. We can assume that Ω is of the form {g ∈ K | ∥1 − g∥ < ε} for some ε, where ∥ · ∥ is the operator norm corresponding to the Euclidean norm on ℝ^{d}. We know that for some n ≤ f_{1} we will find for every γ ∈ Γ^{•}_{n+1} an element β ∈ Γ_{n} such that α(γ) := β^{−1}γ ∈ Ω. Now check conditions a) and b).

Abels 353

References

[1] Abels, H. Discrete groups of affine isolmetries, Journal of Lie Theory 9 (1999), 1–29.

Fakultät für Mathematik Universität Bielefeld Universitätsstrasse 25 D-33501 Bielefeld, Germany abels@mathematik.uni-bielefeld.de

Received March 3, 1999