Consider $Sym\left(n\right)$ endowed with the normalized Hamming metric $d_n$. A finitely generated group $\Gamma$ is P-stable if every almost homomorphism ${\rho }_{n_k}:\Gamma \to Sym\left(n_k\right)$ (i.e., for every $g,h\in \Gamma$, ${lim\hspace{0.17em}}_{k\to \infty }{d}_{n_k}\left({\rho }_{n_k}\right(gh),{\rho }_{n_k}(g\left){\rho }_{n_k}\right(h\left)\right)=0$) is close to an actual homomorphism ${\phi }_{n_k}:\Gamma \to Sym\left(n_k\right)$. Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva and Păunescu showed the same for abelian groups and raised many questions, especially about the P-stability of amenable groups. We develop P-stability in general and, in particular, for amenable groups. Our main tool is the theory of invariant random subgroups, which enables us to give a characterization of P-stability among amenable groups and to deduce the stability and instability of various families of amenable groups.

This article describes the geometry of isomorphisms between complements of geometrically irreducible closed curves in the affine plane ${\mathbb{A}}^2$, over an arbitrary field, which do not extend to an automorphism of ${\mathbb{A}}^2$. We show that such isomorphisms are quite exceptional. In particular, they occur only when both curves are isomorphic to open subsets of the affine line ${\mathbb{A}}^1$, with the same number of complement points, over any field extension of the ground field. Moreover, the isomorphism is uniquely determined by one of the curves, up to left composition with an automorphism of ${\mathbb{A}}^2$, except in the case where the curve is isomorphic to the affine line ${\mathbb{A}}^1$ or to the punctured line ${\mathbb{A}}^1\setminus \left\{0\right\}$. If one curve is isomorphic to ${\mathbb{A}}^1$, then both curves are equivalent to lines. In addition, for any positive integer $n$, we construct a sequence of $n$ pairwise nonequivalent closed embeddings of ${\mathbb{A}}^1\setminus \left\{0\right\}$ with isomorphic complements. In characteristic $0$ we even construct infinite sequences with this property.

Finally, we give a geometric construction that produces a large family of examples of nonisomorphic geometrically irreducible closed curves in ${\mathbb{A}}^2$ that have isomorphic complements, answering negatively the complement problem posed by Hanspeter Kraft. This also gives a negative answer to the holomorphic version of this problem in any dimension $n\ge 2$. The question had been raised by Pierre-Marie Poloni.

We show that many toric domains $X$ in ${\mathbb{R}}^4$ admit symplectic embeddings $\varphi$ into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes $\varphi \left(X\right)$ to $X$. For instance $X$ can be taken equal to a polydisk $P(1,1)$ or to any convex toric domain that both is contained in $P(1,1)$ and properly contains a ball $B^4\left(1\right)$; by contrast a result of McDuff shows that $B^4\left(1\right)$ (or indeed any $4$-dimensional ellipsoid) cannot have this property. The embeddings are constructed based on recent advances in symplectic embeddings of ellipsoids, though in some cases a more elementary construction is possible. The fact that the embeddings are knotted is proved using filtered positive $S^1$-equivariant symplectic homology.

We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of ${\mathbf{PGL}}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus.

Using the new invariants we significantly strengthen results toward the equidistribution of packets of periodic torus orbits on higher rank $S$-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adèlic torus and are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer. The proof combines geometric invariant theory, Galois actions, local arithmetic estimates, and homogeneous dynamics.