Geometric Combinatorics

Abstract

I like to use the term geometric combinatorics to describe those areas of mathe- matics which study combinatorial objects possessing some geometric flavor.Exam- ples of such objects include Projective Planes.Buildings.Geometries in the sense of Tits.Certain graphs.Abstract simplicial complexes.All these examples can be subsumed within the last two examples, which are closely related.Thus I will concentrate on graphs and simplicial complexes.To provide some focus, I will talk about one particular application of geometric combinatorics in finite group theory.But I hope this discussion will illuminate some fairly general concepts.In the application I will discuss, one seeks to prove the uniqueness of the sporadic groups subject to the hypothesis that the group possesses certain subgroups.The technique can also be used to establish presentations for the group.The approach involves associating to the group a graph or simplicial complex.The key step is to show the graph or complex is simply connected.Now some details.We begin with some examples.Examples (1) Let $\Gamma$ be a (undirected) graph.The clique complex of $\Gamma$ is the simplicial complex $K(\Gamma)$ with vertex set $\Gamma$ and simplices the finite cliques of F. Coversely if $L$ is a simplicial complex the graph $\Delta(L)$ of $L$ is the graph whose vertices are the vertices of $L$ and with $x$ adjacent to $y$ if $\{x, y\}$ is a simplex of $L$ .Notice $L$ is a subcomplex of $K(\triangle(L))$ and $\Gamma=\triangle(K(\Gamma))$ .(2) Let $G$ be a group, $\mathcal{F}=(G_{i} : i\in I)$ a finite family of subgroups of $G$ , and $C(G, \mathcal{F})$ the simplicial complex with vertex set $1I_{i}^{G}/G_{i}$ and with simplices the sets $s$ of vertices such that $\bigcap_{C\in s}C\neq\emptyset$ .The complex $C(G, \mathcal{F})$ is the coset complex of $G$ and $\mathcal{F}$ .Notice $G$ is represented as a group of automorphisms on $C(G, \mathcal{F})$ by right multiplication.(3) Let $H$ be a subgroup of $G$ and $\theta$ a selfpaired orbital of $G$ on $G/H$ .Then the graph of $H,$ $\theta$ is the graph with vertex set $G/H$ and edge set $\theta$ .Again $G$ is represented as a group of automorphisms on this graph by right multiplication.

Locations

• IAS/Park City mathematics series - View
• Kyoto University Research Information Repository (Kyoto University) - View - PDF