Given a finite subgroup $\Gamma \subset {\mathrm{SL}}_2\left(\mathbb{C}\right)$ we define an additive $2$-category ${\mathcal{H}}_{\Gamma }$ whose Grothendieck group is isomorphic to an integral form ${\mathfrak{h}}_{\Gamma }$ of the Heisenberg algebra. We construct an action of ${\mathcal{H}}_{\Gamma }$ on derived categories of coherent sheaves on Hilbert schemes of points on the minimal resolutions $\widehat{{ℂ}^2/\Gamma }$.

We consider several ways to measure the “geometric complexity” of an embedding from a simplicial complex into Euclidean space. One of these is a version of “thickness,” based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved for graphs. We also consider the distortion of knots. We give an alternate proof of a theorem of Pardon that there are isotopy classes of knots requiring arbitrarily large distortion. This proof is based on the expander-like properties of arithmetic hyperbolic manifolds.

Let $k$ be a field finitely generated over $\mathbb{Q}$, and let $X$ be a smooth, separated, and geometrically connected curve over $k$. Fix a prime $\ell$. A representation $\rho :{\pi }_1\left(X\right)\to {GL}_m\left({\mathbb{Z}}_{\ell }\right)$ is said to be geometrically Lie perfect if the Lie algebra of $\rho \left({\pi }_1\right(X_{\overline{k}}\left)\right)$ is perfect. Typical examples of such representations are those arising from the action of ${\pi }_1\left(X\right)$ on the generic $\ell$-adic Tate module $T_{\ell }\left(A_{\eta }\right)$ of an abelian scheme $A$ over $X$ or, more generally, from the action of ${\pi }_1\left(X\right)$ on the $\ell$-adic étale cohomology groups $H_{\mathrm{ét}}^i(Y_{\overline{\eta }},{\mathbb{Q}}_{\ell })$, $i\ge 0$, of the geometric generic fiber of a smooth proper scheme $Y$ over $X$. Let $G$ denote the image of $\rho$. Any $k$-rational point $x$ on $X$ induces a splitting $x:{\Gamma }_k:={\pi }_1(Spec(k\left)\right)\to {\pi }_1\left(X\right)$ of the canonical restriction epimorphism ${\pi }_1\left(X\right)\to {\Gamma }_k$ so one can define the closed subgroup $G_x:=\rho \circ x\left({\Gamma }_k\right)\subset G$. The main result of this paper is the following uniform open image theorem. Under the above assumptions, for every geometrically Lie perfect representation $\rho :{\pi }_1\left(X\right)\to {GL}_m\left({\mathbb{Z}}_{\ell }\right)$, the set $X_{\rho }$ of all $x\in X\left(k\right)$ such that $G_x$ is not open in $G$ is finite and there exists an integer $B_{\rho }\ge 1$ such that $[G:G_x]\le B_{\rho }$ for every $x\in X\left(k\right)\setminus X_{\rho }$.

We show that any nonabelian free group $\mathbb{F}$ is strongly ${\aleph }_0$-homogeneous, that is, that finite tuples of elements which satisfy the same first-order properties are in the same orbit under $Aut\left(\mathbb{F}\right)$. We give a characterization of elements in finitely generated groups which have the same first-order properties as a primitive element of the free group. We deduce as a consequence that most hyperbolic surface groups are not strongly ${\aleph }_0$-homogeneous.