5 problems

Let $M$ be a hyperbolic 3-manifold and let $\rho:G\to \operatorname{SL}(2,\mathbb C)$ be a lift of the discrete faithful representation associated to its hyperbolic structure. Supp…

Reduction conjecture. The reduction of $\mu_X^{EP}$ to $\Sigma$ is also an Entov–Polterovich quasimorphism: there exists an Entov–Polterovich quasimorphism

Homogenized signature conjecture. For every positive braid $\beta$,

Let $G$ be a finitely presented group, and let $g_n$ be obtained by a length-$n$ random walk, conditioned to lie in $[G,G]$. Suppose that, for every $\epsilon>0$, there is a positi…

Let $G$ be any finitely generated group, let $S$ be a finite symmetric generating set, and suppose that the space $Q(G)$ of homogeneous quasimorphisms modulo homomorphisms is finit…