Groups of positive weighted deficiency and their applications

TL;DR

This paper introduces weighted deficiency for groups, generalizes classical results, and constructs new types of infinite groups with specific properties, including residually finite and just-infinite groups, with applications in Galois theory.

Contribution

It develops the concept of weighted deficiency and applies it to construct novel infinite groups with prescribed properties, extending classical group theory results.

Findings

Constructed residually finite p-torsion groups with specific subgroup structures

Developed a new method for constructing just-infinite groups with exponential growth

Proved every positive weighted deficiency group has a hereditarily just-infinite quotient

Abstract

In this paper we introduce the concept of weighted deficiency for abstract and pro-$p$ groups and study groups of positive weighted deficiency which generalize Golod-Shafarevich groups. In order to study weighted deficiency we introduce weighted versions of the notions of rank for groups and index for subgroups and establish weighted analogues of several classical results in combinatorial group theory, including the Schreier index formula. Two main applications of groups of positive weighted deficiency are given. First we construct infinite finitely generated residually finite $p$-torsion groups in which every finitely generated subgroup is either finite or of finite index -- these groups can be thought of as residually finite analogues of Tarski monsters. Second we develop a new method for constructing just-infinite groups (abstract or pro-$p$) with prescribed properties; in particular, we show that graded group algebras of just-infinite groups can have exponential growth. We also prove that every group of positive weighted deficiency has a hereditarily just-infinite quotient. This disproves a conjecture of Boston on the structure of quotients of certain Galois groups and solves Problem~15.18 from Kourovka notebook.