2 problems

Let $\Gamma$ be a countable group and let $[\,\cdot\,]\colon\Gamma\to\mathbb{Z}$ be a homomorphism. Suppose that … where $M>\sum_{s\in I}f_s$, and $[s]\geqslant1$ and $f_s>0$ for e…

Let $G$ be a countable group with non-vanishing first $\ell^2$-cohomology. Equivalently, suppose that there is a function $f:G\to\mathbb{C}$ with $f(0)=0$, which is not in…