Consider a finitely generated group that is relatively hyperbolic with respect to a family of subgroups . We present an axiomatic approach to the problem of extending metric properties from the subgroups to the full group . We use this to show that both (weak) finite decomposition complexity and straight finite decomposition complexity are extendable properties. We also discuss the equivalence of two notions of straight finite decomposition complexity.
"Extending properties to relatively hyperbolic groups." Kyoto J. Math. 59 (2) 343 - 356, June 2019. https://doi.org/10.1215/21562261-2018-0017