Block pro-fusion systems for profinite groups and blocks with infinite dihedral defect groups

TL;DR

This work develops block pro-fusion systems for blocks of profinite groups, extending Puig's nilpotent block theory to the profinite setting and relating finite-block fusion data to profinite pro-fusion systems via inverse limits. It proves a profinite version of Puig's structure theorem for nilpotent blocks with topologically finitely generated defect groups, yielding Morita equivalences with $k[[D]]$, and applies this to blocks with the infinite dihedral pro-$2$ defect $D_{2^{\infty}}$, establishing a unique Morita class $k[[D_{2^{\infty}}]]$. The paper also analyzes inverse limits of tame blocks, showing they give bounded completed path algebras with explicit relations, and provides a detailed treatment of dihedral-defect blocks, including a classification up to Morita equivalence in the profinite setting. Finally, it discusses alternative Brauer-pair frameworks and open questions regarding Brauer pairs in profinite groups and extensions beyond countably based $G$.

Abstract

We introduce block pro-fusion systems for blocks of profinite groups, prove a profinite version of Puig's structure theorem for nilpotent blocks, and use it to show that there is only one Morita equivalence class of blocks having the infinite dihedral pro-$2$ group as their defect group.

Theorems & Definitions (52)

• Theorem 1.1: see Theorem \ref{['thm puig in body']}
• Theorem 1.2: see Corollary \ref{['Corollary Dinfty block is kDinfty in body']}
• Proposition 2.1
• proof
• Definition 2.2
• Proposition 2.3: MacQSymInPrep
• Corollary 2.4
• proof
• Proposition 2.5
• proof