We establish integral analogues of results of Bushnell and Henniart for spaces of Whittaker functions arising from the groups ${\mathrm{GL}}_n\left(F\right)$ for $F$ a $p$-adic field. We apply the resulting theory to the existence of representations arising from the conjectural “local Langlands correspondence in families” and reduce the question of the existence of such representations to a natural conjecture relating the integral Bernstein center of ${\mathrm{GL}}_n\left(F\right)$ to the deformation theory of Galois representations.

Let $M$ be an arithmetic hyperbolic $3$-manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of $M$, where each basis element is represented by a surface of “low” genus, and we give evidence for this. We explain the relationship between this conjecture and the study of torsion homology growth.

We prove a Hitchin–Kobayashi correspondence for vortices on the complex affine line with Kähler target, which generalizes a result of Taubes for the case of a line target. More precisely, suppose that $K$ is a compact Lie group and that the target $X$ is either a compact Kähler $K$-Hamiltonian manifold or $X$ is a symplectic vector space with linear $K$-action and a proper moment map. Suppose that the action of the complexified Lie group $G$ satisfies stable $=$ semistable. Then, for some sufficiently divisible integer $n$, there is a bijection between gauge equivalence classes of $K$-vortices with target $X$ and isomorphism classes of maps from the weighted projective line $\mathbb{P}(1,n)$ to $X/G$ that map the stacky point at infinity $\mathbb{P}\left(n\right)$ to the semistable locus of $X$. The results allow the construction and partial computation of the quantum Kirwan map from Woodward and play a role in the conjectures of Dimofte, Gukov, and Hollands relating vortex counts to knot invariants.

Let $V$ be a finite graph, and let $\varphi :V\to V$ be an irreducible train track map whose mapping torus has word-hyperbolic fundamental group $G$. Then $G$ acts freely and cocompactly on a $CAT\left(0\right)$ cube complex. Hence, if $F$ is a finite-rank free group and if $\Phi :F\to F$ is an irreducible monomorphism so that $G=F{\ast }_{\Phi }$ is word-hyperbolic, then $G$ acts freely and cocompactly on a $CAT\left(0\right)$ cube complex. This holds, in particular, if $\Phi$ is an irreducible automorphism with $G=F{⋊}_{\Phi }\mathbb{Z}$ word-hyperbolic.