20 problems

Let $\Gamma$ be an infinite group with Kazhdan property (T), and let $(\Gamma_n)$ be a normal chain in $\Gamma$ with trivial intersection. Write $\mathrm{RG}(\Gamma,(\Gamma_n))$ fo…

Let $G=\prod_{i=1}^{m}\mathbf{G}_{i}(k_{i})$, where each $k_i$ is a local field and each $\mathbf{G}_{i}(k_i)$ is the group of $k_i$-points of a Zariski connected simple $k_i$-alge…

Property $T$ integrality conjecture. The representation $\varrho$ is integral.

Let $X$ be a connected $2$-dimensional simplicial complex whose fundamental group has property T. The finite-cover topological overlap conjecture. There exists $c>0$ such that ever…

Let $G$ be an icc property (T) group. For a $\mathrm{II}_1$ factor $\mathcal M$, let $\mathcal F_s(\mathcal M)$ be the set of all $t\in\mathbb R_+$ for which there exists a $*$-emb…

Let $G$ be an ICC property (T) group, let $H$ be an arbitrary countable ICC group, and let $d:G\times G\to\mathbb{T}$ and $c:H\times H\to\mathbb{T}$ be $2$-cocycles. For $t>0$, sup…

Let $G$ be a semisimple Lie group of rank at least $2$, and let a p.m.p. action of $G$ be irreducible, meaning that every factor of $G$ acts ergodically. An action is essentially f…

Let $k$ be a number field and let $\mathbf{G}$ be a connected, simply connected almost simple $k$-group. Theorem and Corollary assume that every local group $\mathbf{G}(k_s)$ has p…

Let $F$ be a local field of characteristic $0$, let $\mathbf{G}$ be an almost simple, connected and simply connected $F$-algebraic group, and set $G=\mathbf{G}(F)$. In Theorem, whe…

Let $G$ be a connected semisimple Lie group with finite center and without compact factors, and let $\tau$ denote the property T assumption in Theorem. Higher-rank replacement conj…

Spectral conjecture. There exists such a function $f$ and such representations and vectors for every compact $K\subset\Gamma$ and every $\varepsilon>0$. This would yield, by a limi…

Let $G$ be an ICC group with property (T), let $H$ be a group, and let $\operatorname{L}(G)$ and $\operatorname{L}(H)$ denote their group von Neumann algebras. An isomorphism…

Let $G$ be an ICC group with Kazhdan's property (T). Write $\operatorname{Char}(G)$ for the group of characters of $G$, $\operatorname{Out}(G)$ for its outer automorphism group, an…

Asymptotic threshold conjecture. The limit

Admissibility conjecture. A Banach space is admissible if and only if it has nontrivial type.

Let $K$ be a compact group, and let $\Lambda_1$ and $\Lambda_2$ be finitely generated subgroups dense in $K$. Lubotzky–Weiss conjecture. If $\Lambda_1$ is amenable and $\Lambda_2$…

Let $G$ be a countable group with Kazhdan property (T). LERF conjecture. The group $G$ is LERF if and only if it is finite. The preceding proposition establishes the corresponding…

Let $\Phi$ be an irreducible classical root system of rank at least $2$, and let $\Gamma_s(\Phi)$ be its small Weyl graph. The diameter of a graph is the maximum distance between t…

Let $\Gamma=\Gamma_1\times\Gamma_2$, where an action is irreducible when each factor acts ergodically on the underlying measure space. Assume that $\Gamma_1$ has property $(T)$ and…

Let $\Gamma$ be a group with property $(T)$ acting smoothly on a compact manifold $M$ and preserving volume. Let $d(\Gamma)$ be the smallest dimension in which $\Gamma$ has an infi…