Seminarium Zakładu Geometrii

A group G has Serre's property FA if every action of G on a tree has a global fixed point. It is well known that groups with property (T) have property FA. There is a natural generalization of propery FA to higher dimensional actions on simpliclal complexes, called property FA_r where one requires that any action on an r-dimensional CAT(0) simplicial complex has a global fixed point. Farb proved that SL_n(ℤ) enjoys property FA_n-1, but it remains elusive for other higher rank lattices. In a joint work with Ben Lowe, we develop a new method for establishing property FA_r, using minimal submanifolds and min-max theory. As a consequence we show that any cocompact lattice in SL_n(ℝ) has property FA_[n/8] and lattices in the exceptional rank one group F₄^-20 have property FA₂.