Metric ultraproducts of classical groups

Simple non-discrete metric ultraproducts of classical groups are geodesic spaces with respect to a natural metric.


Introduction.
Metric ultraproducts of groups equipped with bi-invariant metrics have been studied in a number of recent papers; see [2,[4][5][6][7]. We recall that a metric or pseudometric d on a group G is called bi-invariant if d(gx, gy) = d(xg, yg) = d(x, y) for all x, y, g ∈ G. We fix an infinite index set I and an ultrafilter U on I. Let (G i , d i ) i∈I be a family of groups with bi-invariant pseudometrics and with diameters sup{d i (g i , 1) | g i ∈ G} bounded independently of i. The Cartesian product i G i of this family has a pseudometric defined by it is a metric group with the metric d induced by the pseudometric on the Cartesian product.
The paper [7] was concerned primarily with the case when each group G i is a non-abelian finite simple group and G is a simple group for which the metric is not discrete. Arguments in that paper show that the group G is then (isomorphic to) a metric ultraproduct of alternating groups with their Hamming metrics, or an ultraproduct of simple classical groups of unbounded rank with natural metrics. It was also shown that if the index set I is countable, then G is path-connected, and, in some cases, is a geodesic space.
For the simple classical groups, there are several natural choices of metric. We shall work with a metric d cr , the classical rank metric, defined below. We shall see (in Lemma 3.1) that d cr is asymptotically equivalent on the family of simple classical groups to the metric d pr used in [7] and some other earlier papers, and so gives rise to the same metric ultraproduct with an equivalent metric.
We recall that a geodesic from one point x of a metric space (X, d) to another point y is an isometric embedding p : [0, d(x, y)] → X such that p(0) = x and p(d(x, y)) = y, and that X is called a geodesic space if all pairs of points are connected by geodesics. Thus geodesic spaces are in particular pathconnected. The path-connectedness of metric ultraproducts of classical groups of unbounded ranks was established in [7]; however the authors were unable to prove in general that they are geodesic spaces with respect to a natural metric. Here we prove the following result.

Word metrics.
We shall use the following result. Proof. For the reader's convenience we include a proof of this particular case of a result of Wantiez [8, p. 147]. It suffices to prove that if g ∈ G \ {1}, then there is a geodesic from 1 to g. Let λ = d (1, g), and let (g i ) ∈ G i be a family that represents g. Thus n −1 i w i (g i ) → U λ. We write each g i as a product of w i (g i ) elements of X i ∪ X −1 i . For each r ∈ [0, λ] let p i (r) be the product of the first λ −1 rw i (g i ) terms in the product for g i , and let p(r) be the image of (p i (r)) in G. Then p is a geodesic from 1 to g.
We pause briefly to discuss the ultraproducts of alternating groups A n with n 5. The alternating groups carry the normalized Hamming metric d H defined by Thus nd H (g, h) − 1 is equal to the word length of g −1 h corresponding to the generating set of S n consisting of the transpositions. Consider also the word metric d w with respect to the set X of 3-cycles as generating set for A n . Each even permutation that moves precisely r elements can be written as a product of at most r/2 elements of X, but not as a product of fewer than r/3 elements of X.
Vol. 109 (2017) Metric ultraproducts of classical groups 409 that on each family (A ni ) of alternating groups with (n i ) unbounded the metrics n −1 i d w and d H are asymptotically equivalent, and so the corresponding ultraproducts coincide. The hypotheses of Lemma 2.1 evidently hold, and so we have the following assertion.

Proposition 2.2.
Let G be an ultraproduct of alternating groups of unbounded ranks with respect to their Hamming metrics. Then G is path-connected.
In [7] it was proved that the above metric ultraproduct is in fact a geodesic space with respect to the metric induced by the Hamming metrics.
3. Metrics on simple classical groups. Throughout this section, V is an ndimensional vector space over a field F , where either n 3 or n = 2 and |F | 4, and G is the derived group of G, where G is the subgroup of GL(V ) preserving one of the following: (1) the zero form on V , (2) a non-singular symplectic form, (3) a non-singular hermitian form with respect to a nontrivial involution on F , or (4) for char F odd, a non-singular symmetric form, and for char F even, a non-singular quadratic form. The centre Z of GL(V ) consists of the scalar multiplicationsλ by elements λ of F × = F \ {0}. Each finite simple classical group arises as the group G/(Z ∩ G) for one of the above cases. The properties of classical groups that we use can be found in [3,Chapter 2].