# Does almost every pair of elements in a compact Lie group generates the connected component?

It is known that almost every pair of elements in a connected compact Lie group (topologically) generates the group.

Obviously this isn't true for non-connected groups but

Given a compact Lie group $G$, is it true that almost every pair of elements of $G$ generates a subgroup containing the connected component $G^\circ$ of 1?

• No need to vote for close, without leaving the OP time to clarify the question. It's clear that "the connected component" meant "the connected component of 1". – YCor Mar 18 '17 at 1:54
• @IgorRivin: Is it trivial? Note that the question is (currently) about almost every pair of elements of $G$, not of $G^\circ$. Anyway, if the closed subgroup generated by $x,y$ contains any connected component of $G$, then it contains $G^\circ$. – Nate Eldredge Mar 18 '17 at 3:11
• The infinite dihedral group $O_2({\bf R})$ is a famous counterexample to "close enough to Haar measure": All elements of the non-identity component are involutions! But it still does not refute the OP's conjecture. – Noam D. Elkies Mar 18 '17 at 5:21
• @IgorRivin this is why when you think a question is trivial, writing a comment and waiting for answers can be useful before voting to close. It often occurs to me too. (Btw you should retract your closing vote.) – YCor Mar 19 '17 at 2:40
• @IgorRivin The only possibly unclear thing was whether some subgroup "contains the connected component of $G$". Since a subgroup of any topological group contains a connected component iff it contains the connected component of 1, this leaves little ambiguity. Anyway you have still left your closing vote. – YCor Mar 19 '17 at 14:23

The closed subgroups of $G$ not containing the identity component lie in countably many conjugacy classes of subgroups. So it is sufficient to show that for each closed subgroup $H$ not containing the identity component, the probability that Haar-random $g_1$ and $g_2$ both lie in some conjugate of $H$ vanishes.

Such pairs are parameterized by the manifold of triples $x \in G/H$, $g_1 \in x H x^{-1}, g_2 \in x H x^{-1}$, which is a manifold of dimension $(\dim G - \dim H) + 2 \dim H = \dim G + \dim H$.

The image of this manifold in $G \times G$ under the projection $(x,g_1,g_2)\mapsto (g_1,g_2)$ must have measure $0$, as it is the image of a smaller-dimensional manifold (as $H$ does not contain the identity we have $\dim H < \dim G$) under a smooth map (by Sard's theorem).

For $H$ a subgroup, the same argument just barely fails to show that the probability that a single $g$ is contained in a conjugate of $H$ vanishes. Indeed in this case both manifolds have dimension $\dim(G)$. This is convenient as that statement is false, because we could take $H$ to be a maximal torus, or alternately the subgroup generated by a reflection in an infinite dihedral group, as in Noam's example.

• What theorem do you use for image of positive dimension under smooth maps to have measure zero? For instance say for closed manifolds $A,B$ under the smooth projection $A\times B\to A$, the image of $A\times\{b\}$ is all of $A$ even if $A$ is a positive codimension submanifold. – YCor Mar 21 '17 at 23:50
• @Ycor I think he meant that the dimension of $G/H \times H \times H$ is lower than the dimension of $G \times G$, so the image is null by Sard's theorem. – Jack Mar 22 '17 at 0:19
• @Jack Let $M$ be the set of triples $(x,g_1,g_2)$ as I describe. Then the projection map to $G \times G$ from $M$ clearly exists. I claim $M$ is a manifold of dimension $2 \dim H+ (\dim G-\dim H)$. The key point is that locally on $G/H$, we can choose a section $s: G/H \to G$ that defines a coset representation, and then $(xH, h_1,h_2) \to (xH, s(x) h_1 s(x)^{-1}, s(x) h_2 s(x)^{-1})$ is perfectly well-defined. – Will Sawin Mar 22 '17 at 0:35
• The last paragraph was more clear before the revision. – R. van Dobben de Bruyn Mar 22 '17 at 3:37
• @WillSawin I see, so you use this local section to show the set you defined is a submanifold, is that right? (Oh and by the way - you accidentally wrote "as $H$ does no contain the identity..." instead of "identity component"). – Jack Mar 22 '17 at 6:08

In fact, a much stronger result (due to Ito and Kawada), see Theorem 2.3 in Emmanuel Breuillard's notes. To wit, if the support of a measure is not contained in a proper closed subgroup then a random walk is eventually equidistributed. If it is contained in a proper closed subgroup, there are two cases: the first is that the subgroup is of codimension zero (in which case it contains the identity component, and we are done), or it is of positive codimension (which is obviously non-generic).

• @user44191 and Igor no you not only misread Breuillard's assumption but modify it in your comment. No Breuillard's notes are not confusing, but you seem to be confused anyway and this comment in your post is confusing. It's not "generated by neighborhoods of 1" (which is unclear and could be interpreted as "generated by any neighborhood of 1" but "generated by a compact neighborhood of 1", which is also known as "compactly generated", meaning that some compact neighborhood of 1 generates $G$, and is trivially fulfilled if $G$ is any compact group. – YCor Mar 19 '17 at 14:28