Overview
This study investigates the large-scale geometry of specific complexes associated with free groups and general groups. Specifically, it focuses on the relative free splitting complex and the relative free factor complex. The primary finding establishes that these complexes exhibit the property of hyperbolicity.
Research Context
The research is set within the mathematical domain of geometric group theory, focusing on the structural properties of groups and their associated geometric complexes. The subject groups include the rank $n$ free group $F_n$ and, more broadly, a general group $\Gamma$. Key structures under examination are the free splitting complex and the free factor complex. These complexes are considered 'relative' to the choice of a free factor system within the respective group. Understanding the large-scale geometry of such complexes, particularly properties like hyperbolicity, provides insights into the group's structure.
Approach
The approach involves a generalized proof structure. The hyperbolicity of the relative free splitting complex and the relative free factor complex, initially demonstrated for the rank $n$ free group $F_n$, is subsequently presented within a broader context. This broad context allows for the extension of the hyperbolicity finding to these complexes when constructed for a general group $\Gamma$, relative to a chosen free factor system for $\Gamma$. The methodology employed in the proof is indicated to yield specific information regarding 'coarsely transitive families of quasigeodesics' within these complexes. This information is characterized as being expressed through 'fold paths of free splittings'.
Findings
- The relative free splitting complex of the rank $n$ free group $F_n$, relative to a chosen free factor system of $F_n$, is hyperbolic.
- The relative free factor complex of the rank $n$ free group $F_n$, relative to a chosen free factor system of $F_n$, is hyperbolic.
- The hyperbolicity proof is applicable in a general context, establishing the hyperbolicity of the relative free splitting complex of a general group $\Gamma$, relative to a choice of a free factor system of $\Gamma$.
- Similarly, the proof demonstrates the hyperbolicity of the relative free factor complex of a general group $\Gamma$, relative to a choice of a free factor system of $\Gamma$.
- The proof also provides information concerning coarsely transitive families of quasigeodesics within these complexes.
- This information about quasigeodesics is expressed in terms of fold paths of free splittings.