Connectedness Properties of Some Subcomplexes of Twin Buildings and Their Group Theoretic Applications 46 views

In this document, we develop a filtration of one half of a twin building generalizing that of Abramenko and Abels to a setting that does not require a group action. This enables us to develop a sufficient condition for determining when opposition complexes in one half of a twin building are k-connected. From there we apply a criterion of Abramenko that allows us to rephrase this sufficient condition solely in terms of the thickness of the building and the types of its links. From there, we apply a criterion from Gandini and a well-known positive F-criterion to determine upper and lower bounds on the finiteness lengths of parabolic subgroups of groups that act strongly transitively on these twin buildings. When we consider groups with RGD systems, we can translate the building-theoretic conditions into group theoretic terms. In particular this yields group-theoretic criteria that lets us re-obtain known results such as the finiteness length of SL_m(F_q[t]) as well as criteria for computing bounds on the finiteness length of parabolic subgroups of Kac-Moody groups over finite fields.

PHD (Doctor of Philosophy)

Group theory; Algebraic groups; Building theory; Kac-Moody groups

English

2025-07-26