We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model ${\mathbf{Q}}_G$ of semi-infinite flag manifolds, and we prove the Pieri–Chevalley formula, which describes the product, in the $K$-theory of ${\mathbf{Q}}_G$, of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over ${\mathbf{Q}}_G$. In order to achieve this, we provide a number of fundamental results on ${\mathbf{Q}}_G$ and its Schubert subvarieties including the Borel–Weil–Bott theory, whose precise shape was conjectured by Braverman and Finkelberg in 2014.

One more ingredient of this article besides the geometric results above is (a combinatorial version of) standard monomial theory for level-zero extremal weight modules over quantum affine algebras, which is described in terms of semi-infinite Lakshmibai–Seshadri paths. In fact, in our Pieri–Chevalley formula, the positivity of structure coefficients is proved by giving an explicit representation-theoretic meaning through semi-infinite Lakshmibai–Seshadri paths.

The aim of this article is to extend a theorem by Cheeger and Müller to spaces with isolated conical singularities by generalizing the proof of Bismut and Zhang to the singular setting. The main tools in this approach are the Witten deformation and local index techniques.

We study actions of countable discrete groups which are amenable in the sense that there exists a mean on $X$ which is invariant under the action of $G$. Assuming that $G$ is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first ${\ell }^2$-Betti number of $G$ with that of the stabilizer subgroups. In addition, for any marked finitely generated nonamenable group $G$ we establish a uniform isoperimetric threshold for Schreier graphs $G/H$ of $G$, beyond which the group $H$ is necessarily weakly normal in $G$.

Even more can be said in the particular case of an atomless mean for the conjugation action—that is, when $G$ is inner amenable. We show that inner amenable groups have fixed price $1$, and we establish cocycle superrigidity for the Bernoulli shift of any nonamenable inner amenable group. In addition, we provide a concrete structure theorem for inner amenable linear groups over an arbitrary field.

As a special case of inner amenability, we consider groups which are stable in the sense of Jones and Schmidt, obtaining a complete characterization of linear groups which are stable. Our analysis of stability leads to many new examples of stable groups; notably, all nontrivial countable subgroups of the group $H\left(\mathbb{R}\right)$, of piecewise-projective homeomorphisms of the line, are stable.