# The Segal Conjecture for Infinite Groups

## TL;DR

This paper extends the Segal Conjecture to infinite groups, establishing an isomorphism between stable cohomotopy groups under specific geometric conditions, thus broadening its applicability beyond finite groups.

## Contribution

It formulates and proves a version of the Segal Conjecture for infinite groups with finite models for their classifying spaces, generalizing the original conjecture.

## Key findings

- For hyperbolic groups, the zero-th stable cohomotopy of BG matches the I-adic completion of the equivariant stable cohomotopy of underline{E}G.
- The conjecture holds for groups with finite models for their classifying space for proper actions.
- The result connects stable cohomotopy of classifying spaces with algebraic completions in a broader group context.

## Abstract

We formulate and prove a version of the Segal Conjecture for infinite groups. For finite groups it reduces to the original version. The condition that G is finite is replaced in our setting by the assumption that there exists a finite model for the classifying space underline{E}G for proper actions. This assumption is satisfied for instance for word hyperbolic groups or cocompact discrete subgroups of Lie groups with finitely many path components. As a consequence we get for such groups G that the zero-th stable cohomotopy of the classifying space BG is isomorphic to the I-adic completion of the ring given by the zero-th equivariant stable cohomotopy of underline{E}G for I the augmentation ideal.