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Lifting graph automorphisms along solvable regular covers

Haimiao Chen, Jin Ho Kwak

TL;DR

It is shown that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of asolvable cover can be decomosed into layer of lifts of the automorphisms in the layers of the abelIAN covers.

Abstract

A {\em solvable} cover of a graph is a regular cover whose covering transformation group is solvable. In this paper, we show that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of a solvable cover can be decomposed into layers of lifts of the automorphism in the layers of the abelian covers. This procedure is applied to classify metacyclic covers of the tetrahedron branched at face-centers.

Lifting graph automorphisms along solvable regular covers

TL;DR

It is shown that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of asolvable cover can be decomosed into layer of lifts of the automorphisms in the layers of the abelIAN covers.

Abstract

A {\em solvable} cover of a graph is a regular cover whose covering transformation group is solvable. In this paper, we show that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of a solvable cover can be decomposed into layers of lifts of the automorphism in the layers of the abelian covers. This procedure is applied to classify metacyclic covers of the tetrahedron branched at face-centers.

Paper Structure

This paper contains 10 sections, 10 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of the theory of lifts along abelian covers
  3. Lifting along nonabelian covers
  4. Decomposing a regular $A$-cover
  5. Lifting along solvable covers
  6. Application to metacyclic covers of the tetrahedron branched at face-centers
  7. Preparation
  8. $H^{{\rm ab}}$-covers
  9. $[H,H]$-covers of $\Gamma\times_{\varphi}H^{\rm ab}$
  10. The result

Key Result

Proposition 1.1

An isomorphism $\alpha:\Gamma_{1}\to\Gamma_{2}$ can be lifted to $\Gamma_{1}\times_{\phi_{1}}A\to\Gamma_{2}\times_{\phi_{2}}A$ if and only if there exists an automorphism $\sigma:A\to A$ such that the following diagram commutes: \xymatrix{ \pi_{1}(\Gamma_{1},u_{0})\ar[d]_{\phi_{1}}\ar[r]^{\alpha_{\a

Figures (1)

  • Figure 1: The graph $\Gamma$

Theorems & Definitions (17)

  • Proposition 1.1
  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • Remark 3.6
  • Lemma 4.1
  • ...and 7 more