Seminario Teoría de Grupos
Weakly branch actions and Boston’s conjecture
Ponente: Jorge Fariña Asategui (Lund University / Universidad del País Vasco)Fecha: jueves 19 de febrero de 2026 - 11:30Lugar: Aula Naranja, ICMAT
Resumen:
Absolute Galois groups have been studied via p-adic Galois representations with great success in the last decades. However, the Fontaine-Mazur conjecture, a central conjecture in arithmetic geometry, predicts that p-adic Galois representations of maximal unramified pro-p extensions have finite image, and thus are not the appropriate objects to study these extensions. In the late 1990s, Boston proposed considering arboreal representations instead, i.e. actions of a Galois group on a spherically homogeneous rooted tree. Motivated by Serre’s open image theorem and the (incomplete) classification of just-infinite pro-p groups he conjectured that arboreal representations have positive-dimensional images belonging to the class of branch groups, introduced right at the time by Grigorchuk. In fact, he posed the purely group-theoretic conjecture that for just-infinite pro-p groups it is equivalent to be branch and to admit a positive-dimensional action on the p-adic tree.
Boston’s conjecture has been open for a quarter of a century. Recently, the author provided the first examples of just-infinite branch pro-p groups with zero Hausdorff dimension, suggesting potential counterexamples to Boston’s conjecture. However, showing that they are indeed counterexamples has proven to be much more difficult. Motivated by Boston’s conjecture, we conduct a thorough study of the rigidity of the branch and weakly branch actions of a group on a spherically homogeneous rooted tree, a problem first considered by Grigorchuk and Wilson in the early 2000s. In this talk, we shall discuss the rigidity of branch and weakly branch actions and their role on the disproval of Boston’s conjecture.
