$p$-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 distance holds an important role as it connects the combinatorics of lattices and the geometry of root systems. in particular, the fixed-point sets of moy-prasad subgroups are precisely the simplicial balls. in this talk, i'll explain those findings and compute their simplicial volume under certain conditions.

[pre-talk at 1:20pm]