Undecidability of epimorphisms onto products of hyperbolic groups

TL;DR

This paper demonstrates the undecidability of certain algebraic problems involving hyperbolic groups, showing no algorithm can determine specific quotient structures in these complex groups.

Contribution

It constructs explicit examples of hyperbolic groups and subgroups where algorithmic detection of quotient structures is impossible, advancing understanding of computational limits in geometric group theory.

Findings

No algorithm can detect whether a finitely presented subgroup is a quotient of a given product of hyperbolic groups.

Constructs a recursive sequence of hyperbolic groups with undecidable quotient properties.

Shows undecidability in the context of torsion-free, linear, hyperbolic groups mapping onto free groups.

Abstract

We exhibit examples of finitely presented subgroups $P$ of direct products of hyperbolic groups for which there is no algorithm that detects whether a finitely presented group has a quotient isomorphic to $P$. For any torsion-free, linear, hyperbolic group $Q$ that maps onto the free group of rank $2$ and $m\geq 2$, we construct a recursive sequence $(\Gamma_n)_{n\in \mathbb{N}}$ of torsion-free, hyperbolic $C'(\frac{1}{6})$ small cancellation groups, with the property that there is no algorithm determining the values $n\in \mathbb{N}$ such that $\Gamma_n$ has a quotient isomorphic to the direct product $Q^{m}$ of $m$-copies of $Q$.