5 problems

Let $Y$ be a $c$-step nilpotent space, and let $\operatorname{V\delta}^{n}_{Y}(L)$ denote its volume of nullhomotopies in degree $n$ at scale $L$. General nilpotent upper-bound con…

Let $M$ be a square matrix of size at least $m$ with nonzero determinant, let $k<m$, and write $\Gamma_M=\mathbb{Z}^m\rtimes_M\mathbb{Z}$. The subgroup $\mathbb{Z}^m$ is $k$-volume…

Let $M$ be a matrix with at least one eigenvalue off the unit circle, and consider the semidirect product $\mathbb{Z}^k\rtimes_M\mathbb{Z}$. Let $N$ be the diagonal matrix whose di…

Let $\mathcal{H}^3$ and $\mathcal{H}^5$ denote the three- and five-dimensional Heisenberg groups. The area distortion of $\mathcal{H}^3$ in $\mathcal{H}^5$ measures the filling are…

Let $M\in GL(k,\mathbb{Z})$ and let $k\geq 3$. Consider the semidirect product $\mathbb{Z}^k\rtimes_M\mathbb{Z}$, where $M$ acts on $\mathbb{Z}^k$. The subgroup $\mathbb{Z}^k$ is a…