p-adic representations are important objects in number theory, and stable lattices serve as a connection between the study of ordinary and modular representations. These stable lattices can be understood as stable vertices in Bruhat-Tits buildings. From this viewpoint, the study of fixed point sets in these buildings can aid research on p-adic representations. The simplicial balls, in particular, hold an important role as they possess the most symmetry and fastest growth, and are closely related to the Moy-Prasad filtrations. In this talk, I'll explain those new findings, provide a characterization of such simplicial balls, and compute their simplicial volume under certain conditions.