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The Complexity of Topological Group Isomorphism

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The Complexity of Topological Group Isomorphism THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM ALEXANDER S. KECHRIS, ANDRE´ NIES AND KATRIN TENT Abstract. We study the complexity of the isomorphism relation for various classes of closed subgroups of S1. We use the setting of Borel reducibility between equivalence relations on Polish spaces. For profi- nite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound. 1. Introduction Let S1 denote the Polish group of permutations of !. The closed sub- groups of S1 form the points in a standard Polish space. It is well-known that the closed subgroups of S1 (or equivalently, the non-Archimedean Pol- ish groups) are, up to topological group isomorphism, the automorphism groups of countable structures (in a countable language). In some inter- esting cases, model theory can be used to understand natural subclasses. Consider for instance the oligomorphic groups, the groups such that for each n there are only finitely many n-orbits. They are precisely the auto- morphism groups of !-categorical structures. Under this correspondence, topological isomorphism turns into bi{interpretability of the structures [1]. Our main goal is to determine the complexity of the topological isomor- phism relation for various classes of closed subgroups of S1 within the set- ting of Borel reducibility between equivalence relations. See [4] for back- ground on this setting. An important question about an equivalence rela- tion E on a standard Borel space X is whether E is classifiable by countable structures. This would mean that one can in a Borel way assign to x 2 X a countable structure Mx in a fixed countable language in such a way that ∼ xEy , Mx = My. We consider the Borel classes of compact (i.e., profinite) groups, locally compact groups, and oligomorphic groups. We also include the class of Roelcke precompact groups, which generalise both the compact and the oligomorphic groups. We introduce a general criterion to show that each isomorphism relation is classifiable by countable structures. Our proof that the criterion works has two different versions, one on the descriptive set theory side (we reduce to conjugacy of closed subgroups of S1, which implies classifiability by countable structures using a known result from descriptive set theory), the other on the model theoretic side (we directly construct the countable structure in a fixed finite language from the group). Rosendal and Zielinski [10, Prop. 10 and 11] established classifiability by countable structures independently from us and published their result on arXiv in Oct. 2016. Their methods are different from ours: they establish The first author was partially supported by NSF grant DMS 1464475. The second author was partially supported by the Marsden fund of New Zealand. 1 2 ALEXANDER S. KECHRIS, ANDRE´ NIES AND KATRIN TENT the results as corollaries to their theory of classification by compact metric structures. As a second result we conversely provide a Borel reduction of graph iso- morphism to isomorphism of profinite groups, using a topological extension of an argument by Mekler [7]. For oligomorphic groups, it is clear that the identity on R is a lower bound; however we leave open the question whether this lower bound can be improved. We note that for !-categorical structures in a finite language, the bi-interpretability relation is Borel and has count- able equivalence classes, so that the upper bound of graph isomorphism for the corresponding automorphism groups is not sharp. Using Lemma 2.1 below, it is not hard to verify that the isomorphism relation for general closed subgroups of S1 is analytical. It is unknown what the exact complexity of this relation is in terms of Borel reducibility. By the above, graph isomorphism is a lower bound. 2. Preliminaries Effros structure of a Polish space. Given a Polish space X, let F(X) denote the set of closed subsets of X. The Effros structure on X is the Borel space consisting of F(X) together with the σ-algebra generated by the sets CU = fD 2 F(X): D \ U 6= ;g, for open U ⊆ X. Clearly it suffices to take all the sets U in a countable basis hUiii2! of X. The inclusion relation on F(X) is Borel because for C; D 2 F(X) we have C ⊆ D $ 8i 2 N [C \ Ui 6= ;! D \ Ui 6= ;]. The following fact will be used frequently. Lemma 2.1 (see [6], Thm. 12.13). Given a Polish space X, there is a Borel map f : F(X) −! X! such that for a non-empty set G 2 F(X), the image G ! f(G) is a sequence (pi )i2! in X that is dense in G. The Effros structure of S1. For a Polish group G, we have a Borel action G y F(G) given by left translation. In this paper we will only consider the case that G = S1. In the following σ; τ; ρ denote injective maps on initial segments of the integers, that is, tuples of integers without repetitions. Let [σ] denote the set of permutations extending σ: [σ] = ff 2 S1 : σ ≺ fg (this is often denoted Nσ in the literature). The sets [σ] form a base of S1. For f 2 S1 let f n be the initial segment of f of length n. Note that 0 0 the [f n] form a basis of neighbourhoods of f. Given σ; σ let σ ◦ σ be the composition as far as it is defined; for instance, (7; 4; 3; 1; 0)◦(3; 4; 6) = (1; 0i. Definition 2.2. For n ≥ 0, let τn denote the function τ defined on f0; : : : ; ng such that τ(i) = i for each i ≤ n. Definition 2.3. For P 2 F(S1), by TP we denote the tree describing P as a closed set in the sense that [TP ]\S1 = P . Note that TP = fσ : P 2 C[σ]g. Lemma 2.4. The relation f(A; B; C): AB ⊆ Cg on F(S1) is Borel. Proof. AB ⊆ C is equivalent to the Borel condition 8β 2 TB8α 2 TA [jαj > max β ! α ◦ β 2 TC ]. THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM 3 For the nontrivial implication, suppose the condition holds. Given f 2 A; g 2 B and n 2 N, let β = g n, and α = f 1+max β. Since α ◦ β 2 TC , the neighbourhood [f ◦ g n] intersects C. As C is closed and n was arbitrary, we conclude that f ◦ g 2 C. The Borel space of non-Archimedean groups. Lemma 2.5. The closed subgroups of S1 form a Borel set U(S1) in F(S1). Proof. D 2 F(S1) is a subgroup iff the following three conditions hold: • D 2 C[(0;1;:::;n−1i] for each n • D 2 C[σ] ! D 2 C[σ−1] for all σ • D 2 C[σ] \ C[τ] ! D 2 C[τ◦σ] for all σ; τ. It now suffices to observe that all three conditions are Borel. Note that U(S1) is a standard Borel space. The statement of the lemma actually holds for each Polish group in place of S1. Locally compact closed subgroups of S1. These groups are exactly the (separable) totally disconnected locally compact groups. The class of such groups has been widely studied. A set D 2 F(S1) is compact iff the tree TD = fσ : D 2 C[σ]g is finite at each level. A closed subgroup G of S1 is locally compact iff some point in G has a compact neighbourhood. Equivalently there is τ such that G 2 C[τ] and the tree fσ τ : G 2 C[σ]g is finite at each level. Thus, compactness and local compactness of subgroups are Borel conditions in F(S1). The canonical structure for a closed subgroup of S1. Given G 2 U(S1) we can in a Borel way obtain a countable structure MG in a count- ∼ able signature such that G = Aut(MG). For each n, order the n-tuples lexicographically. Let ha i , where k ≤ !, be the ascending list of the i i<kn n n-tuples that are least in their orbits. The signature has n-ary predicate n n symbols Pi for i < kn, where the symbol Pi is interpreted as the G-orbit of ai. 3. A sufficient criterion for classifiability of topological isomorphism by countable structures We show that the isomorphism relation on a Borel class V of subgroups of S1 that is invariant under conjugacy is classifiable by countable structures, provided that we can canonically assign to each G 2 V a countable base of neighbourhoods of 1 that are open subgroups. For instance, in the case of locally compact groups G, we can take as NG the compact open subgroups of G. This relies on the classic result of van Dantzig: if a totally disconnected group G is locally compact, it has a compact open subgroup U. Note that U is profinite, so in fact 1 has a basis of neighbourhoods consisting of compact open subgroups. 4 ALEXANDER S. KECHRIS, ANDRE´ NIES AND KATRIN TENT 3.1. The sufficient criterion. Theorem 3.1. Let V be a Borel set of subgroups of S1 that is closed under conjugation. Suppose that for each G 2 V we have a countably infinite set NG of open subgroups of G that forms a neighbourhood basis of 1. Suppose further that the relation T = f(G; U): G 2 V and U 2 NGg is Borel, as well as isomorphism invariant in the sense that ∼ φ: G = H implies V 2 NG $ φ(V ) 2 NH .
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