39 problems

Connes' rigidity conjecture. If

Bader–Furman–Gelander–Monod's conjecture. Higher-rank algebraic groups should possess Banach property (T) with respect to the class of all super-reflexive Banach spaces.

Let $C$ be a compact group containing finitely generated dense subgroups $G$ and $H$, where $G$ is amenable and $H$ has property $(\operatorname{T})$. Lubotzky–Weiss conjecture. Th…

Kazhdan property (T) conjecture. In the general case, this order is conjectured to be correct; equivalently, the relevant automorphism groups of free groups should have Kazhdan's p…

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 $G_1$ and $G_2$ be ICC groups with property (T), and let $\mathcal{L}(G_i)$ denote their group von Neumann algebras. Connes' conjecture. Within the class of ICC property (T) gr…

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…

Let $d$ be a positive integer, and let the universal lattices be the groups $\mathrm{SL}_d(\mathbb{Z}[x_1,\dots,x_k])$. Shalom's conjecture. Many properties of the groups…

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

Let $n>1$, let $M$ be a property (T) subfactor of an ultrapower $(L\mathbb F_n)^{\mathcal U}$, and suppose that the relative commutant $M'\cap (L\mathbb F_n)^{\mathcal U}$ is a fac…

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, let $\mathcal{L}(G)$ denote its group von Neumann algebra, and let $\mathscr{I}(\mathcal{L}(G))$ be the set of Jones indices of finite-index s…

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…

A group $G$ is framenable if it has the framenability property defined in the paper, and a group has property (T) in the sense of Kazhdan. Framenability versus property (T) conject…

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 $G$ be an ICC property (T) group, and let $\operatorname{L}(G)$ be its group von Neumann algebra. Denote by $\mathcal F_{\mathrm{s}}(\operatorname{L}(G))$ the standard fundamen…

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…

Popa's conjecture. Every $\mathrm{II}_1$ factor with property (T) has trivial fundamental group:

Let $G$ be a connected Lie group with property (T). For a complete finite-volume Riemannian manifold $M$, its Cheeger constant is … where the infimum is over compact smooth submani…

Asymptotic threshold conjecture. The limit