Unbounded representations of groups

Question

Let $H$ be a Hilbert space and $G$ be a finitely generated group. Let $\pi:G\rightarrow GL(H)$ be a representation.

A map $c:G\rightarrow H$ is called cocycle if $c(gh)=π(g)c(h)+c(g)$ for all $g,h$ in the group $G$. A cocycle $c$ is proper if the map $g\mapsto c(g)$ is proper, i.e., for every constant $K$ the number of elements $g$ in the group such that $||c(g)|| \lt K$ is finite.

Q1: Does there always exists a representation (bounded or not) of a group on a Hilbert space which admits a proper cocycle?

the answer to this question is in the comment of Mikael.

Q2: Does there always exists a representation (by bounded operators) of a group on a Hilbert space which admits a proper cocycle?

Here we only assume that $||\pi(g)||<\infty$, but $\pi$ is not necessarily uniformly bounded.

Answer 1

Q1 and Q2 have a positive answer for all countable groups (conversely a discrete uncountable group cannot bear any proper function). Let $\mu$ be a proper function from $G$ to the positive reals, and view it as a discrete measure on $G$. Assume in addition that $\mu$ grows reasonably, and more precisely satisfies an equality of the form $\mu(gh)\le C_g\mu(h)$ with $C_G\>0$ (e.g. fix a proper subadditive length $|\cdot|$ and define $\mu(g)=|g|+1)$.

Then the action of left action of $G$ on itself induces a well-defined left regular representation $\pi$ of $G$ on $\ell^2(G,\mu)$, which is bounded ($\|\pi(g)\|\le C_g^{1/2}$).

Let $e$ be the unit in $G$ and $\delta_g$ the Dirac function at $g\in G$. Define $b$ as the coboundary $b(g)=\delta_e-g\delta_e=\delta_e-\delta_g$. Then $\|b(g)\|\ge \mu(g)-\|\delta_e\|$, which is proper; thus $b$ is a proper cocycle.