10 problems

Subexponential regulator conjecture.

Let $M^n$ be a closed aspherical $n$-manifold with residually finite fundamental group, and set $G=\pi_1(M^n)$. Let $G_k\mathrel{\lhd}G$ be any normal chain of finite-index subgrou…

Suppose that … is a tower of covers of congruence arithmetic hyperbolic 3-manifolds whose injectivity radius approaches infinity. Write $H_1(M_i,\mathbb{Z})_{tors}$ for the torsion…

Let $F_m$ be a finitely generated free group, let $\varphi$ be an automorphism, and set … Assume that $\varphi$ is polynomially growing of degree $d$. For a Farber sequence…

Let $G$) be an infinite residually finite $2$-acyclic group of type $VF$. Write $\rho^{(2)}(G)$ for its $2$-torsion and $\rho^\mathbb{Z}(G)$ for its integral torsion. Lück's conjec…

Torsion-homology conjecture. For every prime number $p$, there exist strongly isospectral closed hyperbolic $3$-manifolds $M_1$ and $M_2$ such that

Let $Y(K_{\Sigma})$ and $Y'(K_{\Sigma})$ be a Jacquet–Langlands pair, let $\mathbf{T}$ be the relevant Hecke algebra, and let $\mathfrak{m}$ be a non-Eisenstein maximal ideal. Writ…

Torsion conjecture (T1.5). There exists a finite cover $X$ of $T_\psi$ and a universal tower of finite abelian covers $\{X_i\}$ of $X$ which have exponential torsion homology growt…

Torsion conjectures. The manifold $T$ has exponential torsion homology growth with respect to some tower of finite covers. In fact, $T$ has exponential torsion homology growth with…

Let $\Gamma_0({\mathfrak p})$ be a congruence subgroup inside a fixed Bianchi group, where ${\mathfrak p}$ is a prime ideal of residue degree one. Write $N{\mathfrak p}$ for its no…