Marius Dadarlat
Purdue University, West Lafayette, IN, USA

For a countable discrete groups $G$ we consider $\varepsilon$-representations of $G$ into unitary groups $U(n)$. These are unital maps $\rho:G \to U(n)$ such that $\|\rho(st)-\rho(s)\rho(t)\|<\varepsilon$ for all $s,t\in G$. Kazhdan has shown that the surface groups of genus $>1$ admit $\varepsilon$-representations which are far from genuine representations in the point-norm topology.

We exhibit new classes of hyperbolic groups $G$ which have the same instability features. More precisely, there exist a finite subset $F\subset G$ and $C>0$ with the following property. For any $\varepsilon>0$ there is an $\varepsilon$-representation $\rho:G \to U(n)$ such that for any representation $\pi:G\to U(n)$, $\max_{s\in F} \|\rho(s)-\pi(s)\|>C.$