# The stable category and invertible modules for infinite groups

## TL;DR

This paper develops a stable module category for certain infinite groups and computes its Picard group, especially when the group acts on a tree with finite stabilizers, advancing understanding of invertible modules.

## Contribution

It introduces a well-behaved stable category for infinite groups and provides methods to compute the Picard group in specific group actions, a novel approach in the field.

## Key findings

- Constructed a stable module category for a broad class of infinite groups.
- Calculated the Picard group for groups acting on trees with finite stabilisers.
- Enhanced understanding of invertible modules in the context of infinite groups.

## Abstract

We construct a well-behaved stable category of modules for a large class of infinite groups. We then consider its Picard group, which is the group of invertible (or endotrivial) modules. We show how this group can be calculated when the group acts on a tree with finite stabilisers.