Quotients of higher dimensional Cremona groups

We study large groups of birational transformations $\mathrm{Bir}(X)$, where $X$ is a variety of dimension at least $3$, defined over $\mathbb{C}$ or a subfield of $\mathbb{C}$. Two prominent cases are when $X$ is the projective space $\mathbb{P}^n$, in which case $\mathrm{Bir}(X)$ is the Cremona group of rank $n$, or when $X \subset \mathbb{P}^{n+1}$ is a smooth cubic hypersurface. In both cases, and more generally when $X$ is birational to a conic bundle, we produce innitely many distinct group homomorphisms from $\mathrm{Bir}(X)$ to $\mathbb{Z}/2$. As a consequence we also obtain that the Cremona group of rank $n \ge 3$ is not generated by linear and Jonquires elements. Joint work with Stphane Lamy and Susanna Zimmermann.