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, Maru\v{s}i\v{c} 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. In this talk, we provide 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.

张志硕，本科毕业于北京航空航天大学华罗庚班，直博到墨尔本大学，现为墨尔本大学在读博士生。博士导师为夏彬绉和周三明教授，研究方向主要为群与图。已在Forum Math. Sigma，J. Algebra，Comm. Algebra 等期刊发表5篇论文。