Branch groups and new types of subgroup growth for pro-$p$ groups

Yiftach Barnea, Jan-Christoph Schlage-Puchta

Published 2016-11-22Version 1

We show that there are uncountably many different growth types for the subgroup growth of pro-$p$ groups, thus, answering one of the main open problems in the theory of subgroup growth. As a by product we show that a class of pro-$p$ branch groups including the Grigorchuk group and the Gupta-Sidki groups have all subgroup growth type $n^{\log n}$.