Cotlar identities were used as a tool to show the $L_p$ ($1 < p < \infty$) boundedness of Hilbert transforms on the real line. In 2017, Mei and Ricard first generalized Cotlar identities to the non-commutative setting and showed the $L_p$-boundednesses ($1 < p < \infty$) of Hilbert transforms on free groups. Then in 2022, Gonz{\'a}lez-P{\'e}rez, Parcet and Xia generalized the result on groups acting on $\mathbb{R}$-trees by using a geometric model. In this talk, we will further discuss this topic on Coxeter groups (abstract reflection groups) and groups which admit actions on buildings. This is a joint work with Runlian Xia.
