Narrow Results

We show that the rigid $C^{\ast }$-tensor categories of finite-dimensional type 1 unitary representations of the quantum groups $U_q({𝔤}_2)$ corresponding to the exceptional Lie group $G_2$ for positive $q\ne 1$ have property (T).

A compactly generated group is noncompact if and only if it admits a nonconstant harmonic function (for some, equivalently for every, reasonable measure). This generalizes the known fact that a finitely generated group is infinite if and only if it admits a nonconstant harmonic function (for some, equivalently every, reasonable measure).

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following.

• If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group.

• If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an ℝ-tree is trivial, then H is Hopfian.

• If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian.

• Any finitely presented group is isomorphic to a finite index subgroup of Out(H) for some group H with Kazhdan property (T). (This sharpens a result of Ollivier–Wise).

We extend the results of Ershov, Jaikin-Zaiprin, and Kassabov on property (T) to certain groups "coordinatized" by nonassociative algebras, thereby giving new examples of groups with property (T).

In [5], Fox, Gromov, Lafforgue, Naor and Pach, in a response to a question of Gromov [7], constructed bounded degree geometric expanders, namely, simplical complexes having the affine overlapping property. Their explicit constructions are finite quotients of $\widetilde{A_d}$-buildings, for $d\ge 2$, over local fields. In this paper, this result is extended to general high rank Bruhat–Tits buildings.