Chromatic congruences and Bernoulli numbers

TL;DR

This paper establishes new congruences related to orbifold Euler characteristics and Bernoulli numbers, connecting classical results with p-adic limits and applying them to mapping class groups.

Contribution

It introduces novel congruences for orbifold Euler characteristics and Bernoulli numbers, extending classical congruences through p-adic analysis and group theory.

Findings

New congruences for orbifold Euler characteristics

p-adic limits recover Brown-Quillen congruence

Classical Bernoulli number congruences are recovered

Abstract

For every natural number $n$ and a fixed prime $p$, we prove a new congruence for the orbifold Euler characteristic of a group. The $p$-adic limit of these congruences as $n$ tends to infinity recovers the Brown-Quillen congruence. We apply these results to mapping class groups and using the Harer-Zagier formula we obtain a family of congruences for Bernoulli numbers. We show that these congruences in particular recover classical congruences for Bernoulli numbers due to Kummer, Voronoi, Carlitz, and Cohen.