An analytic family of uniformly bounded representations of free groups

For right-angled Coxeter groups 𝑧^{|𝑔|} is a coefficient of a uniformly bounded representation

“…The first inequality comes from the fact that p is an isometric embedding, and the second from an estimate of the norms of representations coming from trees, adapted by Valette from [5]. This proves the theorem.…”

Nonunitarizable Representations and Random Forests

“…Very explicit nonunitarizable representations of F 2 were constructed in the 1980s [8,39,45]. It follows by induction of representations that any group containing F 2 as a subgroup is nonunitarizable.…”

Are Unitarizable Groups Amenable?

“…Here of course G = SL 2 (R) is viewed as a Lie group, but a fortiori the discrete group G d underlying SL 2 (R) fails to be unitarizable, and since every group is a quotient of a free group and "unitarizable" obviously passes to quotients, it follows (implicitly) that there is a non-unitarizable free group, from which it is easy to deduce (since unitarizable passes to subgroups, see Proposition 0.5 below) that F 2 the free group with 2 generators is not unitarizable. In the 80's, many authors, notably Mantero-Zappa [46]- [47], Pytlik-Szwarc [70], Bożejko-Fendler [9], Bożejko [8], . .…”

“…non-unitarizable representations on F 2 or on F ∞ (free group with countably infinitely many generators), see [45] for a synthesis between the Italian approach and the Polish one. See also Valette's papers [78]- [79] for the viewpoint of groups acting on trees, (combining Pimsner [68] and [70]) and [32] for recent work on Coxeter groups. This was partly motivated by the potential applications in Harmonic Analysis of the resulting explicit formulae (see e.g.…”