Linearity of compact analytic groups over domains of characteristic zero

Abstract: A $p$-adic analytic group is a topological group that is endowed with an analytic manifold structure over $\mathbb{Z}_p$, the ring of $p$-adic integers. This definition can be extended by considering the manifold structure over more general pro-$p$ domains, such as the power series rings $\mathbb{Z}_p[[t_1, \dots, t_m]]$ or $\mathbb{F}_p[[t_1, \dots, t_m]]$ (where $\mathbb{F}_p$ denotes the finite field of $p$ elements).

Lazard established already in the 1960s that compact $p$-adic analytic groups are linear, as they can be embedded as a closed subgroup within the group of invertible matrices over $\mathbb{Z}_p$. Nonetheless, the question of the linearity of analytic groups over more general domains remains unsolved.

In this talk, we shed some light to this question by proving that when the coefficient ring is of characteristic zero, every compact analytic group is linear. We will provide background on the problem and outline the strategy of our argument.

Joint with M. Casals-Ruiz.

group theory

Audience: researchers in the topic

Groups in Galway 2024