15 problems

Let $M$ be an aspherical closed manifold. Consider a descending chain of subgroups … such that each $G_i$ is normal in $G=\pi_1(M)$, the index $[G:G_i]$ is finite, and…

Lück–Sauer–Wegner conjecture. Their $L^2$-torsions satisfy

Let $\Gamma$ be a discrete group whose classifying space $B\Gamma$ is a finite CW complex of odd dimension, and let $E\Gamma$ be its universal cover. Let $\phi_{E\Gamma}$ denote th…

Let $M$ and $N$ be compact odd-dimensional manifolds, and let $f:M\to N$ be a homotopy equivalence. Let $\widehat M$ and $\widehat N$ be their normal $\Gamma$-covering spaces, and…

Let $F_1$ and $F_2$ be finitely generated free groups, let $\varphi\colon F_1\to F_2$ be a homomorphism, and let $\tau^{(2)}_u(\varphi)$ denote its universal $L^2$-torsion. Univers…

Let $G$ be a $\det$-finite group, meaning that the relevant Fuglede–Kadison determinants are finite, and let $\Phi\colon G\to G$ be an automorphism. Assume that $G$ has subexponent…

Modified homological torsion approximation conjecture. The limit exists and equals

Lück's continuity conjecture. The function $A^{(2)}(N,\psi,t)$ is continuous with respect to $(\psi,t)\in H^1(N;\mathbb{R})\times\mathbb{R}^+$.

Chain flare conjecture. The chain flare condition holds whenever the monodromy is exponentially growing.

Let $G$ be the reductive group defining the symmetric space $\widetilde{X}$, let $\tau\in\operatorname{Rep}(G(\mathbb{R}))$ satisfy $\tau\not\cong\tau_\theta$, and let…

Let $\Gamma_1$ and $\Gamma_2$ be profinitely commensurable arithmetic subgroups of linear algebraic groups, satisfying the assumptions of the source's main theorem, and suppose tha…

Friedl–Lück–Tillmann's conjecture. If $G$ is amenable but not virtually $\mathbb{Z}$, then

Let $G$ be an infinite amenable group for which $L^2$-torsion is defined. Vanishing conjecture for $L^2$-torsion. The $L^2$-torsion vanishes: … The conjecture concerns the expected…

Let $G$ and $H$ be countable groups such that all the $L^2$-Betti numbers of $G$ and $H$ vanish. Assume that both $G$ and $H$ admit finite CW-models for their classifying spaces. T…

Let $M$ be a closed aspherical manifold of odd dimension and let $\widetilde{M}$ be its universal covering. $L^2$-torsion conjecture. The space $\widetilde{M}$ is determinant-$L^2$…