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Chromatic congruences and Bernoulli numbers

Irakli Patchkoria

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.

Chromatic congruences and Bernoulli numbers

Abstract

For every natural number and a fixed prime , we prove a new congruence for the orbifold Euler characteristic of a group. The -adic limit of these congruences as 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.

Paper Structure

This paper contains 7 sections, 21 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chromatic congruences and the proof of Theorem \ref{['theorema']}
  4. Generating tuples and the proof of Theorem \ref{['theoremb']}
  5. The congruence at height $n=\infty$ and a new proof of the Brown-Quillen congruence
  6. An application: Mapping class groups and congruences for Bernoulli numbers
  7. An Example: $\Gamma_{\frac{(p-1)(p-2)}{2}}$ and a comparison of the chromatic and Brown-Quillen congruences

Key Result

Theorem 1

Let $p$ be a prime and $n \geq 1$. Suppose $G$ is a virtually torsion-free discrete group with a finite $\underline{E}G$. Then we have where $C\langle g_1, \dots, g_n\rangle$ denotes the centralizer of the subgroup generated by the tuple $(g_1, \dots, g_n)$.

Theorems & Definitions (45)

  • Theorem 1
  • Remark
  • Theorem 2
  • Proposition \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 35 more