Applications to Group Theory

14 Applications to Group Theory

14.1 Covering Spaces of a Graph

14.1.1 A linear graph is a space X that is written as the union of a collection of subspaces A_{\alpha}, each of which is an arc, s.t.

14.1.1.1 The intersection A_{\alpha} \cup A_{\beta} of two arcs is either empty or consists of a single point that is an end point of each.

14.1.1.2 The topology of X is coherent with the subspaces A_{\alpha}.

14.1.1.2.1 The arcs A_{\alpha} are called the edges of X, and their interiors are called the open edges of X, their end points are called the vertices of X; we denote the set of vertices of X by X^{0}.

14.1.2 (Lem 83.1) Every linear graph X is Hausdorff; in fact it is normal.

14.1.3 Let X be a linear graph. Let Y be a subspace of X that is a union of edges of X. Then Y is closed in X and is itself a linear graph ; we call it a subgraph of X.

14.1.4 (Lem 83.2) Let X be a linear graph. If C is a compact subspace of X, there exists a finite subgraph Y of X that contains C. IF C is connected, Y can be chosen to be connected.

14.1.5 (Lem 83.3) If X is a linear graph, then X is locally path connected and semilocally simply connected.

14.1.6 (Thm 83.4) Let p : E->X be a covering map, where X i s a linear graph. If A_{\alpha} is an edge of X and B is a path component of p^{-1}(A_{\alpha}), then p maps B homeomorphically onto A_{\alpha}. Furthermore, the space E is a linear graph, with the path componenets of the spaces p^{-1}(A_{\alpha}) as its edges.

14.2 The fundamental group of a graph

14.2.1 An oriented edge e of a graph X is an edge of X together with an ordering of its vertices; the first is called the initial vertex, and the second, the final vertex, of e,. an edge path in X is a sequence e_{1}, …, e_{n} of oriented edges of X s.t. the final vertex of e_{i} equals the initial vertex of e_{i+1}, for i = 1, …, n-1. Such an edge path is entirely specified by the sequence of vertices x_{0}, .., x_{n}, where x_{0} is the initial vertex of e_{1} and x_{i} is the final vertex of e_{i} for i = 1,…,n. It is said to be an edge path from x_{0} to x_{n}. It is called a closed edge path if x_{0} = x_{n}.

14.2.1.1 Given an oriented edge e of X, let f_{e} be the positive linear map of [0,1] onto e; it is a path from the initial point of e to the final point of e. Then, corresponding to the edge path e_{1}, .., e_{n} from x_{0} to x_{n}, one has the actual path f = f_{1} * (f_{2} * (\cdots * f_{n})) from x_{0} to x_{n}, where f_{i} = f_{e_{i}}; it is uniquely determined by the edge path e_{1}, .., e_{n}. We call it the path corresponding to the edge path e_{1}, .., e_{n}. If the edge path is closed, then the corresponding path f is a loop.

14.2.2 (Lem 84.1) A graph X is connected iff every pair of vertices of X can be joined by an edge path in X.

14.2.3 Let e_{1}, …, e_{n} be an edge path in the linear graph X. It can happen that for some i, the oriented edges e_{i} and e_{i+1} consist of the same edge of X, but with opposite orientations. If this situation does not occur, then the edge path is said to be a reduced edge path.

14.2.4 A subgraph T of X is said to be a tree in X if T is connected and T contains no closed reduced edge paths.

14.2.5 (Lem 84.2) If T is a tree in X, and if A is an edge of X that intersects T in a single vertex, then T \cup A is a tree in X. Conversely, if T is a finite tree in X that consists of more than one edge, then there is a tree T_{0} in X and an edge A of X that intersects T_{0} in a single vertex, such that T = T_{0} \cup A.

14.2.6 (Thm 84.3) Any tree T is simply connected.

14.2.7 A tree T in X is maximal if there is no tree in X that properly contains T.

14.2.8 (Thm 84.4) Let X be a connected graph. A tree T in X is maximal iff it contains all the vertices of X.

14.2.9 (Thm 84.5) If X is a linear graph, every tree T_{0} in X is contained in a maximal tree in X.

14.2.10 (Lem 84.6) Suppose X = U \cup V, where U and V are open sets of X. Suppose that U \cap V is the union of two disjoint open path connected sets A and B, that \alpha is path in U from the point a of A to the point b of B, and that \beta is a path in V from b to a. If U and V are simply connected, then the class [\alpha * \beta ] generates \pi_{1}(X,a).

14.2.11 (Thm 84.7) Let X be a connected graph that is not a tree, Then the fundamental group of X is an nontrivial free group, Indeed, if T is a maximal tree in X, then the fundamental group of X has a system of free generators that is in bijective correspondence with the collection of edges of X that are not in T.

14.3 Subgroups of Free groups

14.3.1 (Thm 85.1) If H is a subgroup of a free group F, then H is free.

14.3.2 If X is a finite linear graph, we define the Euler number of X to be the number of vertices of X minus the number of edges. It is commonly denoted by \chi (X).

14.3.3 (Lem 85.2) If X is a finite, connected linear graph, then the cardinality of a system of free generators for the fundamental group of X is 1- \chi (X).

14.3.4 Let H be a subgroup of the group G. If the collection G?H of right cosets of H in G is finite, its cardinality is called the index of H in G.

14.3.4.1 The collection of left cosets of H in G h as the same cardinality as it.

14.3.5 (Thm 85.3) Let F be a gree group with n+1 free generators; let H be a subgroup of F. If H has index k in F, then H has kn+1 free generators.