Speaker: Professor Alexander A. Ivanov，Hebei Normal University

Inviter: Shaoshi Chen

Title: Local Graph Theory for the Higman–Sims Group

Language: English

Time & Venue: 2026.06.15 14:00-15:00 南楼N219

Abstract: During the last quarter of the twentieth century, an approach to proving the existence and uniqueness of sporadic and other simple groups was developed. This approach is based on an ingenious combination of group-theoretical techniques with methods from graph theory and topology. It is known as the Group Amalgam Method. The graph-theoretical aspect of the Group Amalgam Method begins with a graph Γ on which the group G in question acts as a group of automorphisms, with the action being transitive on both vertices and edges. Consequently, for every vertex x of Γ, the isomorphism type of the subgraph ∆ induced by the vertices adjacent to x is independent of the choice of x. It is assumed that ∆ is connected. This local data, although possibly incomplete or even hypothetical, serves as a starting point. The group G can be constructed and proved unique provided that Γ is shown to be the only connected graph which is locally ∆ in the above sense. This leads to the following general problem of Local Graph Theory.

Problem A. For a given graph ∆:

(a) prove or disprove the existence of a graph Γ which is locally ∆;

(b) construct the universal cover of Γ with respect to the local isomorphism;

(c) classify, up to isomorphism, all connected graphs which are locally ∆.