The Finite and Infinite Generation of Groups Acting on CAT(0)-cube complexes

Abstract

This MPhil thesis explores groups acting on CAT(0)-cube complexes X- in particular, non-uniform lattices Γ ≤ Aut(X). The first section provides a new proof of the classical result that non-uniform tree lattices are not finitely generated. Using the more general setting of essential actions studied in detail in Caprace and Sageev’s Rank Rigidity of CAT(0)-cube complexes and Sageev’s early work, the author proves that non-uniform lattices acting on CAT(0)-cube complexes with strict fundamental domain are not finitely generated- generalising the work of Thomas and Wortman in the case of non-uniform lattices of right-angled buildings. The final section is the construction of the quotient and complex of groups of SL2(F2[t, t−1 ]) acting on T3 × T3, the product of Bruhat-Tits trees corresponding to SL2(F2((t))) and SL2(F2((t −1 ))).