74 problems

Strong Atiyah conjecture. If $\operatorname{lcm}(G)$ is finite, then

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…

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 ……

Singer–Hopf conjecture.

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 $G$ be a torsion-free countable group, let $k$ be a subfield of $\mathbb{C}$, and let $\D(k[G])$ be the Linnell ring, defined as the division closure of $k[G]$ in the ring of u…

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

Let $\Gamma$ be a torsion-free abstract group, and let $\operatorname{rk}_{\Gamma}$ denote its associated rank function. The Atiyah conjecture. The value of…

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

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…

Let $G$ be a group and let $A\in\operatorname{Mat}_n(\mathbb{Z}[G])$ be a globally positive matrix, meaning a matrix to which the stated positivity condition applies. Determinant C…

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

Gromov's zero-in-the-spectrum conjecture. There is $p \geq 0$ such that zero belongs to the spectrum of $(\Delta_p)_{\operatorname{min}}$.

Let $W$ be a discrete group, let $L^2W$ be the Hilbert space completion of its real group algebra, and let $R_M:(L^2W)^n\rightarrow(L^2W)^m$ be right multiplication by a matrix $M$…

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

Determinant-class conjecture. Every finite $CW$-complex is of determinant class.

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…

Connes–Shlyakhtenko's conjecture. The quantities $\beta^{(2)}_k(M,\tau)$ should be interesting invariants of the von Neumann algebra $M$. These $L^2$-Betti numbers extend the group…

Gromov's conjecture. For every $p\in\mathbb{N}$,

Let $G$ be a discrete group, let ${\mathcal N}(G)$ be its group von Neumann algebra, and let $\frac{1}{{\mathcal F}{\mathcal I}{\mathcal N}(G)}{\mathbb Z}$ be the additive subgroup…

Mathai's converse conjecture. If $\pi_1(M)$ is torsion-free, then $\rho_{(2)}$ is a homotopy invariant.

Let $M^{4k}$ be an open complete manifold of finite volume and bounded covering geometry outside a compact subset, and let the integral of the Hirzebruch signature form over…

Let $\Gamma$ be a torsion-free group. Consider the maps … and … defined by the corresponding $L^2$-signature and ordinary signature constructions. Torsion-free group-ring signature…

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 $H^n$ be the real hyperbolic space of dimension $n$, and let $T^{(2)}(H^n)$ denote its $L^2$-torsion density, defined using the invariant metric on $H^n$. Nonvanishing conjectu…