Abstract: We will discuss a new result on the p-adic geometry of Igusa stacks, which allows one to relate the p-adic geometry of Shimura varieties for a pair of groups $G, G'$ which are global extended pure inner forms. Using this geometric input and the spectral action of Fargues-Scholze, we will discuss how this input may be used to prove the existence of Hecke-equivariant cohomological "Jacquet-Langlands" maps between the cohomology (even with integral or torsion coefficients) of the two Shimura varieties of abelian type associated to $G, G'$.
