7 problems

Let $G$ be a group with finite presentation $\langle X \mid R \rangle$, and let $p$ be a prime. Write $d_p(G)$ for the dimension of the first homology of $G$ with coefficients in…

Let $k\subset S^3$ be a knot with Alexander polynomial $\Delta$, and let $M_r$ be the $r$-fold cyclic cover of $S^3$ branched over $k$. The finite values of $|H_1(M_r)|$ are those…

Higher-rank homology growth conjecture. For every prime $p$ and every $n\in\\{0,\dots,r-1\\}$,

Let $M$ be a finite volume hyperbolic 3-manifold with fundamental group $G$. For finite-index subgroups $K\leq G$, write $(H_1(K,\mathbb{Z}))_{tor}$ for the torsion subgroup of the…

Let $G$ be a higher-rank real semisimple Lie group, let $X$ be its associated symmetric space, and let $(\Gamma_n)_{n\in\mathbb{N}}$ be a sequence of pairwise non-conjugate lattice…

Let $M$ be a closed, orientable surface of genus $g$ equipped with a generic Finsler metric $m$. Let $d4a2$ be the set of elements of $d4a2$ and, for each $d4a2$, let…

Let $M$ be an aspherical closed manifold of dimension $d$, let $G=\pi_1(M)$, let $\widetilde M$ be its universal cover, and let $(G_i)_{i\in I}$ denote the finite-index subgroups u…