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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 $\pi$ of a group on a Hilbert space which admits a proper cocycle and such that $||\pi(g)||<\infty$?

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.

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

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.

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

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 $\pi$ of a group on a Hilbert space which admits a proper cocycle and such that $||\pi(g)||<\infty$?

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

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 $||c(g)||<K$ is finite.

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

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

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.

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