9 problems

Lubotzky–Sarnak conjecture. For some sequence of finite-index normal subgroups intersecting only at the identity, the Cayley graphs of the corresponding finite quotients do not for…

Let $G$ be a group with finite presentation $\langle X \mid R \rangle$, and let $p$ be a prime. Write $d_p(G)$ for the dimension of the first homology of $G$ with coefficients in…

Let $\mathcal{F}_n$ be the free group of rank $n$, and let $\operatorname{Aut}(\mathcal{F}_n)$ and $\operatorname{Out}(\mathcal{F}_n)$ denote its automorphism and outer automorphis…

Let $G_i$ be the groups in an expander family of Cayley graphs $\mathcal{C}(G_i,S_i)$, and let $\operatorname{Aut}(F_n)$ denote the automorphism group of a free group. Property tau…

The groups $\mathrm{SL}_d(\mathcal{O})$, where $\mathcal{O}$ is the ring of integers in a number field $K$, share several properties, including Kazhdan's property $T$, a positive s…

Let $\Gamma$ be an irreducible higher-rank lattice. Lubotzky's property $\tau$ conjecture. The group $\Gamma$ has property $\tau$. This is listed among consequences of the spectral…

Perfect-closure property $(\tau)$ conjecture. The group $\Gamma$ has property $(\tau)$ with respect to the family $\{\Gamma(I)\}$ as $I$ ranges over all finite-index ideals of $R$.…

Let $R=\mathbb{Z}\langle x,y\rangle$ be the free non-commutative ring on $x$ and $y$. Fix an integer $d\geq 3$, and let $\Gamma=EL_d(R)$ be the group generated by the elementary un…

Let $Gamma$ be a finitely generated subgroup of $GL(n,mathbf C)$ that is not virtually soluble. Property tau conjecture. The stronger assertion proposed by the authors is that…