New cosystolic high-dimensional expanders from KMS groups

Abstract

Cosystolic expansion is a high-dimensional generalization of the Cheeger constant for simplicial complexes. Originally, this notion was motivated by the fact that it implies the topological overlapping property, but more recently it was shown to be connected to problems in theoretical computer science such as list agreement expansion and agreement expansion in the low soundness regime. There are only a few constructions of high-dimensional cosystolic expanders and, in dimension larger than $2$, the only known constructions prior to our work were (co-dimension 1)-skeletons of quotients of affine buildings. In this paper, we give the first coset complex construction of cosystolic expanders for an arbitrary dimension. Our construction is more symmetric and arguably more elementary than the previous constructions relying on quotients of affine buildings. The coset complexes we consider arise from finite quotients of Kac--Moody--Steinberg (KMS) groups and are known as KMS complexes. KMS complexes were introduced in recent work by Grave de Peralta and Valentiner-Branth where it was shown that they are local-spectral expanders. Our result is that KMS complexes, satisfying some minor condition, give rise to infinite families of bounded degree cosystolic expanders of arbitrary dimension and for any finitely generated Abelian coefficient group. This result is achieved by observing that proper links of KMS complexes are joins of opposition complexes in spherical buildings. In order to show that these opposition complexes are coboundary expanders, we develop a new method for constructing cone functions by iteratively adding sets of vertices. Hence we show that the links of KMS complexes are coboundary expanders. Using the prior local-to-global results, we obtain cosystolic expansion for the (co-dimension 1)-skeletons of the KMS complexes.