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A Few Remarks on Invariable Generation in Infinite Groups

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A Few Remarks on Invariable Generation in Infinite Groups A few remarks on invariable generation in infinite groups Gil Goffer Gennady A.Noskov The Weizmann Institute of Science The Sobolev Institute of Mathematics 76100, Rehovot, Israel 644043, Omsk, Russia February 12, 2018 Abstract A subset S of a group G invariably generates G if G is generated by {sg(s)|s ∈ S} for any choice of g(s) ∈ G, s ∈ S. A topological group G is said to be IG if it is invariably generated by some subset S ⊆ G, and TIG if it is topologically invariably generated by some subset S ⊆ G. In this paper we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group SL2(R) and the automorphism group of a regular tree are TIG, and that the groups P SLm(K),m ≥ 2 are not IG for countable fields of infinite transcendence degree over a prime field. Contents 1 Introduction 2 2 Preliminary results 3 3 Parabolics and stabilizers as Wiegold subgroups 6 arXiv:1802.10427v2 [math.GR] 22 Apr 2020 4 Borel versus Wiegold 9 5 Compact Lie groups and their generalizations 10 6 SL2(R) is T IG 10 6.1 Conjugacy classes of SL2(R)................................... 11 6.2 One-parameter subgroups of SL2(R) .............................. 11 6.3 One-dimensional Lie subgroups of SL2(R)andtheirspectra . .. .. .. .. .. .. .. 13 6.4 2-dimensional Lie subgroups of SL2(R)andtheirspectra. .. .. .. .. .. .. .. .. 13 6.5 Mainlemmaandtheorem14 .............................. .... 14 1 7 Aut(T ) is T IG 14 7.1 The classification of isometries . ....... 15 7.2 Vertexstabilizers................................... ...... 15 7.3 Conjugacy classes of Aut(T )................................... 16 7.4 Generatingavertextransitivesubgroup . ........... 17 7.5 Proofoftheorem21 ................................. ...... 17 8 Uncountably many countable non-IG -groups 18 8.1 Lawsingroups ...................................... .... 18 8.2 Rationalpoints ..................................... ..... 19 8.3 Building up free tuples . ..... 19 8.4 When is PSLm (K) an ? ................................... 20 IG 9 Open problems 21 10 Acknowledgements 22 1 Introduction The notion of invariable generation of groups goes back at least as far as Jordan’s paper of 1872. Theorem 1 ([Jo, Se]) . Assume that a finite group G acts transitively on a set X of cardinality at least 2. Then there exists g G displacing every element of X, i.e. g acts fixed-point-freely on X. ∈ Independently, but more than hundred years later, J. Wiegold introduced the class X of groups G with the property that whenever G acts transitively on a set X of cardinality at least 2, there exists an element g G acting fixed-point-freely on X. (See [Wi1]). Thus Jordan’s theorem above says that every finite ∈ group belongs to the class X. Not only Wiegold failed to mention Jordan, but also he claims indirectly that Jordan’s theorem is obvious. “Observe that, in particular, finite groups and soluble groups are in X – obvious facts once noted, though I know of no reference to them in the literature”. (See [Wi1]). Wiegold gave two more characterizations of the class X. Namely, G X if and only if for every proper subgroup ∈ H G the union Hg : g G does not cover the whole G. (Here, we use the common notation Hg ≤ S{ ∈ } to denote the conjugate g−1Hg of H.) Secondly, G X if and only if every subset, containing at least ∈ one representative of every conjugacy class of G, generates G. These characterizations lead naturally to the following definitions (of our own, though the concepts implicitly occurred in above conversation). A subset S of a group G is called conjugation complete (in G) if it meets every non-trivial conjugacy class of G. A proper subgroup W of a group G is called a Wiegold 2 subgroup if G is covered by conjugates of W , i.e. G = W g. This is equivalent to saying that W G Sg∈G ≤ is proper, and W is conjugation complete. In these terms it is easy to see that the class X consists of those groups that have no Wiegold subgroups, or equivalently, groups in which every conjugation complete set generates G. Now Jordan’s theorem becomes really “obvious” for finite groups, by a counting argument. Indeed, arguing by contradiction, suppose that G is a finite group with a Wiegold subgroup W . Then G = W g. There are at most S G : W members in this union, each member is of cardinality W and all members contain the identity | | | | element, hence the cardinality of the union is strictly less than G : W W = G , contradicting that W | | · | | | | is Wiegold. Independently of Wiegold, but somewhat later, J. D. Dixon has invented the notion of invariable generation for finite groups (see [Di]). A subset S of a finite group G invariably generates G if G is generated by sg(s) s S for any choice of g(s) G, s S. This notion was motivated by Chebotarev’s { | ∈ } ∈ ∈ Density Theorem [Ch]. W. M. Kantor, A. Lubotzky and A. Shalev, following Dixon, have defined a (not necessarily finite) group G to be invariably generated if any conjugation complete subset of G generates G (see [KLS2]). They also discovered that the class of invariably generated groups coincides with IG Wiegold’s class X. Since then a few works have been done showing that specific families of groups are invariably generated by given sets, or proving other groups are not by any subset. See the papers IG [Di, Ge, GM, GGJ, KLS1, KLS2, Wi1, Wi2] containing a lot of information on the class . IG In this paper we study the notion of topological invariable generation (abbreviated to ). is T IG T IG a weaker property that , and there are plenty of examples for groups which are but not . For IG T IG IG instance, it is proved in this paper that the Lie group SL (R) is , whereas it seems extremely difficult 2 T IG to determine the status of SL2(R) endowed with the discrete topology. In this paper we study the problem of topological invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group SL2(R) and the group of tree automorphisms Aut(T ) are (sections 6 and 7), and that the groups PSLm(K),m 2 are not T IG ≥ for countable fields of infinite transcendence degree over the prime field (section 8). In section 2 we IG give an alternative description of in terms of Wiegold subgroups and in terms of fixed-point free T IG actions. We clarify the relationship between the notions of Wiegold, Borel and parabolic subgroups in sections 3-5. 2 Preliminary results We shall formulate now the invariable generation problem in a topological setup. An action G y X of a topological group G on a topological space X is continuous if the map (g, x) gx from G X to X is 7→ × continuous. For a subgroup W G and an element g G, let W g = g−1Wg denote the conjugate of W ≤ ∈ 3 by g. A subset S of G is called conjugation complete (in G) if it meets every non-trivial conjugacy class of G. We call a proper closed subgroup W of G a Wiegold subgroup if G is covered by conjugates of W , i.e. G = W g, or equivalently if W G is proper, and W is conjugation complete. g∈G ≤ S α An action G y X of a group G on a set X has fixed-point property (for short, α is an (FP)-action) if every g G has a fixed point in X. Alternatively, α is an FP-action if and only if G is covered by the ∈ point stabilizers, i.e. G = Gx, where the stabilizer of x is Gx = g G : gx = x . Sx∈X { ∈ } Lemma 2 The following conditions on a topological group G are equivalent: I. G contains a Wiegold subgroup. II. There is a continuous transitive FP-action G y X on a Hausdorff topological space X of cardinality at least 2. III. There exists a conjugation complete subset S of G that does not generate G topologically. Proof. I ⇒ II. Suppose that G contains a Wiegold subgroup, say W . The natural action G y (G/W ) is transitive, and since W is proper, the cardinality of G/W is at least 2. Moreover, every g G is −1 −1 ∈ contained in some conjugate W h ,h G. The inclusion g W h is equivalent to the equality ∈ ∈ ghW = hW , which in turn means that g fixes the point hW in the topological quotient G/W . Thus the action G y (G/W ) satisfies the fixed-point property. II ⇒ I. Suppose that II holds for the action G y X. We claim that the stabilizer Gx is Wiegold for every x X. Fix x X. First, Gx is proper since the action is transitive. Next, the ∈ ∈ stabilizer Gx is a closed subgroup of G. This is because, as the action is continuous, the orbit map −1 π : G X : g gx is continuous, and Gx = π ( x ), with x closed. By fixed-point property, → 7→ { } { } G is covered by stabilizers, i.e. G = Gy. And by transitivity every y X is of the form Sy∈X ∈ −1 y = gx for some g G, thus G is covered by stabilizers of the form Ggx = gGxg for g G, and ∈ ∈ so Gx is Wiegold. I ⇒ III. Let W be a Wiegold subgroup of G. Then the subset W of G is conjugation complete, and since W is a proper closed subgroup of G, it does not generate G topologically.
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