Appendix: Boundedly Generated Groups with Pseudocharacter(s)

The aim of this appendix is to construct concrete groups which simultaneously: (1) are boundedly generated; (2) have Kazhdan’s property (T); (3) have a one-dimensional space of pseudocharacters. By (3), such groups don’t have property (QFA), whilst they have property (FA) by (2); moreover the quasimorphisms in (3) cannot be bushy in the sense of [9]. Property (3) has its own interest, as all previous constructions yield infinite-dimensional spaces. (By taking direct products of our examples, one gets any finite dimension.) The examples will be lattices Γ̃ in non-linear simple Lie groups; more precisely, starting with certain higher rank Lie groups H with π1(H) = Z and suitable lattices Γ < H, the group Γ̃ will be the preimage of Γ in the universal covering central extension