Hereditarily just-infinite groups with positive first $L^2$-Betti number

Examples of hereditarily just-infinite, residually finite groups are rare. Even in the well-studied class of finitely generated, residually finite infinite torsion groups, it is difficult to find examples. In this talk, we present a new construction of infinite torsion groups, which enables us to control finite quotients and normal subgroups. As an application, we describe the first examples of residually finite, hereditarily just-infinite groups with positive first $L^2$-Betti number. We will explain how the lower bound for the first $L^2$-Betti number is obtained and we will indicate why these groups have polynomial normal subgroup growth (answering a question of Barnea and Schlage-Puchta). Eventually, we give an outlook to related constructions.

This is based on joint work with Eduard Schesler.