Relative free splitting and free factor complexes: An overview

For any group $\Gamma$ and any free factor system~$\mathscr A$ of $\Gamma$, the relative outer automorphism group $\text{Out}(\Gamma;\mathscr A)$ acts naturally on the relative free splitting complex $\mathcal{F\!S}(\Gamma;\mathscr A)$ and on the complex of relative free factor systems $\mathcal{F\!F}(\Gamma;\mathscr A)$, generalizing the well known actions of $\text{Out}(F_n)$ on the absolute free splitting complex $\mathcal{F\!S}(F_n)$ and the absolute free factor complex ${\mathcal{F}}(F_n)$ of the rank~$n$ free group~$F_n$. This overview summarizes a three part work regarding the large scale geometry of $\mathcal{F\!S}(\Gamma;\mathscr A)$ and $\mathcal{F\!F}(\Gamma;\mathscr A)$ and the geometric dynamics of the actions on these complexes by elements of $\text{Out}(\Gamma;\mathscr A)$. In Part I arXiv:1407.3508 we prove hyperbolicity of $\mathcal{F\!S}(\Gamma;\mathscr A)$ and of $\mathcal{F\!F}(\Gamma;\mathscr A)$. In Parts II and III arXiv:2212.09907, arXiv:2503.07532 we study the relation between the geometric dynamics of an element of $\text{Out}(\Gamma;\mathscr A)$ and the dynamics of its relative train track representatives. The main tool in Part II is the \emph{Two Over All Theorem}, expressing an exponential flaring property of Stallings fold paths in $\mathcal{F\!S}(\Gamma;\mathscr A)$. The main tools in Part III are \emph{filling paths}, used to formulate and prove a strong version of the \emph{Two Over All Theorem}.