Topological generation results for free unitary and orthogonal groups

Abstract

We show that for every N ≥ 3 the free unitary group U + N is topologically generated by its classical counterpart U N and the lower-rank U + N −1 . This allows for a uniform inductive proof that a number of finiteness properties, known to hold for all N = 3, also hold at N = 3. Specifically, all discrete quantum duals U + N and O + N are residually finite, and hence also have the Kirchberg factorization property and are hyperlinear. As another consequence, U + N are topologically generated by U N and their maximal tori Z * N (dual to the free groups on N generators) and similarly, O + N are topologically generated by O N and their tori Z * N 2 .

Affine noncommutative geometry

“…can be proved by recurrence on N. Indeed, at N = 1 there is nothing to prove, at N = 2 this is something well-known, and elementary, as explained for instance in [44], [46], and in general, the recurrence step N − 1 → N can be established as follows:…”

“…We will need the following key result, coming from [39], [44], [46]: Theorem 2.24. The following happen:…”

“…(2) A key result from [44], [46], whose proof is quite technical, not to be explained here, states that we have the following generation formula, valid at any N ≥ 3:…”

Quantum permutation groups

“…(2) B + N , C + N are Fourier liberations too, by using a Fourier transform. (3) S + N is a Fourier liberation too, being generated by its tori [39], [43]. (4) H + N , K + N remain to be investigated, by using the general theory in [78].…”

“…Generally speaking, this is something quite difficult, because for the empty graph itself we are in need of the above-mentioned technical results from [39], [43].…”