47 problems

Lück's homotopy-invariance conjecture. $L^2$-Reidemeister torsion is a homotopy invariant.

Let $G$ be a group, let $\operatorname{Wh}(G)$ be its Whitehead group, and let $A\in GL_n(\mathbb ZG)$. Let $r_A^{(2)}$ be the $G$-equivariant bounded operator on $\ell^2(G)^n$ giv…

Let $G$ be a discrete group. The algebraic eigenvalue property is the property that the eigenvalues under consideration for the combinatorial Laplacian are algebraic numbers. Algeb…

Let $G$ be a finitely generated group satisfying the Atiyah conjecture, and let $C_*$ be a finitely generated free $L^2$-acyclic $\mathbb{Z}G$-chain complex. For a cohomology param…

Let $M$ be a closed aspherical $(2n+1)$-manifold. For a cohomology class $\phi\in H^1(M;\mathbb{R})$, let $b_i^{(2)}(\widetilde M;\phi)$ denote the twisted $L^2$-Betti numbers of t…

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$, and let $G$ be a countable group with a bound on the orders of its finite subgroups. For a finite matrix $A$ over $\mathbb{K}[G]$, l…

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$, let $G$ be a countable group with a bound on the orders of its finite subgroups, and let $\mathrm{lcm}(G)$ denote the least common m…

Let $G$ be the fundamental group of a finite-volume hyperbolic manifold of odd dimension, assume that $G$ is RFRS, and let $\beta_i^{(2)}(G;\mathbb K)$ denote its $i$th $L^2$-Betti…

Let $M$ be a closed aspherical manifold with RFRS fundamental group $G=\pi_1(M)$, and let $\beta_i^{(2)}(G;\mathbb K)$ denote the $i$th $L^2$-Betti number with coefficients in a fi…

Let $G$ be a $\mathrm{PD}^n$-group that is RFRS. Here $\beta_i^{(2)}(G;\mathbb K)$ denotes the $i$th $L^2$-Betti number with coefficients in a field $\mathbb K$. Poincaré-duality-g…

Atiyah conjecture. The von Neumann dimension of the kernel of this map is an integer.

Suppose $\Gamma\overset{\alpha}{\curvearrowright}(X,\mu)$ is a probability-measure-preserving action, with associated invariant random subgroup $H_\alpha$ and orbit equivalence rel…

Let $\Gamma$ be a discrete group, let $L(\Gamma)$ be its group von Neumann algebra, and let $\tau$ be the standard trace on $L(\Gamma)$. For $A\in M_n(\mathbb{Z}[\Gamma])$, write ……

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…

Singer–Hopf conjecture.

Let $\mathsf{G}$ be a semisimple algebraic group over $\mathbb{C}$, let $\Gamma$ be a torsion-free subgroup of type $FP_\infty$, and let $\lambda_i=\lambda_i(k)$ be the highest wei…

Let $\mathsf{G}$ be a semisimple algebraic group over $\mathbb{C}$, let $\Gamma$ be a torsion-free subgroup of type $FP_\infty$, and let $\{\mathsf{W}_k\}$ be a sequence of irreduc…

A group is locally indicable if every nontrivial finitely generated subgroup admits a surjective homomorphism onto $\mathbb{Z}$. The locally indicable subgroup-rigidity conjecture.…

Let $\Gamma$ be a group, let $K$ be a field, and let … be a descending chain of normal subgroups of finite index with trivial intersection. For a matrix $A$ over $K[\Gamma]$, defin…

$L^2$-Hall characterization conjecture. The following conditions should be equivalent:

Finite abelian hierarchy conjecture. Then $G$ should have the $L^2$-Hall property and satisfy the Geometric Hanna Neumann conjecture.

Let $X$ be an aspherical two-complex and let $G=\pi_1(X)$ be its fundamental group. Say that $G$ has trivial second $L^2$-Betti number when $b_2^{(2)}(G)=0$. Betti coherence conjec…

Let $M^n$ be a closed aspherical manifold. Singer's conjecture. The rational $L^2$-Betti numbers of its fundamental group vanish outside the middle dimension: … This is a basic van…

Let $G$ be a group, let $\mathbb C G$ be its complex group algebra, and let $\ell^2(G)$ be the Hilbert-space completion of $\mathbb C G$ in the 2-norm. Linnell's analytic zero divi…

Atiyah's integrality conjecture.