Asymptotic measures and links in simplicial complexes

Abstract

We introduce canonical measures on a locally finite simplicial complex $K$ and study their asymptotic behavior under infinitely many barycentric subdivisions. We also compute the face polynomial of the asymptotic link and dual block of a simplex in the $d^{th}$ barycentric subdivision $S{d}^d(K)$ of $K$, $d\gg 0$. It is almost everywhere constant. When $K$ is finite, we study the limit face polynomial of $S{d}^d(K)$ after F.Brenti-V.Welker and E.Delucchi-A.Pixton-L.Sabalka.

15 pages, 1 figure