Determining the vertex stabilizers of 4-valent half-arc-transitive graphs

TL;DR

The paper addresses the problem of which concentric groups can occur as vertex stabilizers in connected $4$-valent half-arc-transitive graphs, proving that every tightly concentric group is a $4$-HAT-stabilizer and thereby extending the known list of stabilizers. It develops a general framework that treats the auxiliary permutation $x_\tau$ on a concentric group $H$ and studies the generated group $\langle R(H),x_\tau\rangle$, aiming to realize $\mathrm{Alt}(H)$ as the automorphism group in a way that yields a $4$-valent half-arc-transitive action. The main contribution is Theorem: every tightly concentric group is a $4$-HAT-stabilizer, with a constructive proof that also yields the corollary that $\mathcal{H}_7\times C_2^{m-7}$ is a $4$-HAT-stabilizer for all $m\ge7$, expanding the known families and supporting the conjecture that all concentric groups are $4$-HAT-stabilizers. The approach hinges on a four-step program (primitivity, non-affine, non-wreath, Alt$(H)$ identification) combined with the O'Nan-Scott framework, enabling a systematic exclusion of primitive almost-simple and diagonal types and culminating in the Alt$(H)$ realization that yields the stabilizer. This advances the broader program of classifying vertex stabilizers for half-arc-transitive graphs and points toward a full conjectural characterization that every concentric group is a $4$-HAT-stabilizer.

Abstract

We say that a group is a $4$-HAT-stabilizer if it is the vertex stabilizer of some connected $4$-valent half-arc-transitive graph. In 2001, Marušič and Nedela proved that every $4$-HAT-stabilizer must be a concentric group. However, over the past two decades, only a very small proportion of concentric groups have been shown to be $4$-HAT-stabilizers. This paper develops a theory that provides a general framework for determining whether a concentric group is a $4$-HAT-stabilizer. With this approach, we significantly extend the known list of $4$-HAT-stabilizers. As a corollary, we confirm that $\mathcal{H}_7\times C_2^{m-7}$ are $4$-HAT-stabilizers for $m\geq 7$, achieving the goal of a conjecture posed by Spiga and Xia.

Figures (1)

• Figure 1: Illustration of concentric groups

Theorems & Definitions (70)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Conjecture 1.8
• Lemma 2.1
• proof