Strong algebraization of fixed point properties

TL;DR

This paper provides an intrinsic criterion for synthesizing relative fixed point properties into the whole fixed point property for certain Busemann NPC spaces, advancing the construction of super-expanders from finite simple groups of Lie type.

Contribution

It introduces a new criterion that synthesizes fixed point properties without Bounded Generation assumptions, addressing a question posed by Shalom.

Findings

Established an intrinsic criterion for fixed point properties

Applied the criterion to Busemann NPC spaces

Suggested a pathway for constructing super-expanders from finite simple groups

Abstract

The following natural question arises from Shalom's innovational work (1999, Publ. IHES): "Can we establish an intrinsic criterion to synthesize relative fixed point properties into the whole fixed point property without assuming Bounded Generation?" This paper resolves this question in the affirmative. Our criterion works for ones with respect to certain classes of Busemann NPC spaces. It, moreover, suggests a further step toward constructing super-expanders from finite simple groups of Lie type.