On the Haagerup and Kazhdan properties of R. Thompson’s groups

A machinery developed by the second author produces a rich family of unitary representations of the Thompson groups F, T and V. We use it to give direct proofs of two previously known results. First, we exhibit a unitary representation of V that has an almost invariant vector but no nonzero {[F,F]} -invariant vectors reproving and extending Reznikoff’s result that any intermediate subgroup between the commutator subgroup of F and V does not have Kazhdan’s property (T) (though Reznikoff proved it for subgroups of T). Second, we construct a one parameter family interpolating between the trivial and the left regular representations of V. We exhibit a net of coefficients for those representations which vanish at infinity on T and converge to 1 thus reproving that T has the Haagerup property after Farley who further proved that V has this property.

Diffeomorphism groups of tame Cantor sets and Thompson-like groups

The group of ${\mathcal{C}}^{1}$-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson’s groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin’s higher-dimensional generalizations $nV$ of Thompson’s group $V$ arise when we consider products of central ternary Cantor sets. We derive that the ${\mathcal{C}}^{2}$-smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.

Office Hours with a Geometric Group Theorist

Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. This book brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors. An essential primer for undergraduates making the leap to graduate work, the book begins with free groups—actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples. It features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.

Growth of positive words and lower bounds of the growth rate for Thompson’s groups F(p)

Let [Formula: see text], [Formula: see text] be the family of generalized Thompson’s groups. Here, [Formula: see text] is the famous Richard Thompson’s group usually denoted by [Formula: see text]. We find the growth rate of the monoid of positive words in [Formula: see text] and show that it does not exceed [Formula: see text]. Also, we describe new normal forms for elements of [Formula: see text] and, using these forms, we find a lower bound for the growth rate of [Formula: see text] in its natural generators. This lower bound asymptotically equals [Formula: see text] for large values of [Formula: see text].