On Subgroups of Quasi-Graph Groups

A graph is called a quasi-graph if the case of an edge of the graph equals its inverse is allowed. A graph of groups is called a quasi-graph of groups if the corresponding graph is a quasi-graph. A group is termed quasi-graph group if it is a fundamental group of a non-trivial quasi-graph of groups. In this paper we show that a subgroup of a quasi-graph group is a quasi-graph group. Mathematics Subject Classification: 20E08, 20F05, 20E06


Introduction
In [4], Mahmood introduced the concepts of quasi-graphs of groups and their fundamental groups.The main result of this paper is to show that if G is a fundamental group of a quasi-graph of groups and if H is a subgroup of G, then we use the results of [5] and [6] to show that H is a fundamental group of a quasigraph of groups.This paper is divided in to 4 sections.In section 2, we introduce the concepts quasi graphs, groups acting on trees with inversions, and their fundamental domains.In section 3, we use the results of section 2 to obtain the structures of quasi-graphs of groups induced by the fundamental domains for groups acting on trees with inversions.In section 4, we apply the results of section 3 to obtain the structures of quasi-graphs of groups induced by the subgroups of the fundamental groups of quasi-graphs of groups.

Quasi-Graphs Induced by Fundamental Domains
A quasi graph X consists of two disjoint sets V(X), (the set of vertices of X) and E(X), (the set of edges of X), with V(X) non-empty, together with three functions 0  : E(X)V(X), 1  : E(X)V(X), and  :E(X)E(X) is an involution satisfying the conditions that  (e) = t(e), and  (e) = e .This implies that o( e ) = t(e), t( e ) = o(e), and e = e.The case e = e is allowed.There are obvious definitions of subgraphs, circuits, trees, morphisms of graphs and Aut(X), the set of all automorphisms of the graph X which is a group under the composition of morphisms of graphs.For more details, the interested readers are referred to in [1], [2] and [9].We say that a group G acts on a graph X if there is a group homomorphism : G Aut(X).In this case, if xX (vertex or edge) and gG, we write g(x) for ( (g))(x).Thus, if gG, and yE(X), then g(o(y)) = o(g(y)), g(t(y)) = t(g(y)), and g( y ) = ) ( y g . The case the action with inversion is allowed. That is; g(y) = ( y ) is allowed for some gG, and yE(X).In this case we say that g is an inverter element of G and y is called an inverted edge.
If the group G acts on the graph X and xX, (x is a vertex or edge), then 1.The stabilizer of x, denoted x G is defined to be the set x G ={gG: g(x) = x}.It is clear that x G  G, and if xE(X), and u{o(x), t(x)}, then 2. The orbit of x is the set G(x) = {g(x): gG}.It is clear that G acts on the graph X without inversions if and only if G( e ) ≠ G(e) for any eE(X).

The set of orbits is denoted by
forms a quasi-graph.Definition.Let G be a group acting on a connected quasi-graph X with inversions, and let T and Y be two subtrees of X such that TY, and each edge of Y has at least one end in T. Assume that T and Y are satisfying the following.(i) T contains exactly one vertex from each vertex orbit.(ii) Y contains exactly one edge y(say) from edge orbit if G(y) ≠ G( y ) and exactly one pair x, x from each edge orbit if G(x) = G( x ).Then (1) T is called a tree of representatives for the action of G on X, (2) Y is called a transversal for the action of G on X.For simplicity we say that (T; Y) is a fundamental domain for the action of G on X.For more details, the readers are referred to [3].The properties of fundamental domains for the actions of groups on connected quasi-graphs imply that if G is a group acting on a connected quasi-graph X with inversions, and (T; Y) is a fundamental domain for the action of G on X, then for any vV(Y) and any y E(Y), there exists a unique vertex denoted v* of T such that g(v*) = v; that is, G(v*) = G(v), and element denoted [y] of G satisfying the follows.The main result of this section is the following lemma.Lemma 2.1.Let G be a group acting on a connected quasi-graph X with inversions, and let (T; Y) be a fundamental domain for the action of G on X.

Let
)} ( : ≠ .The fact that for vV(X) and eE(X), the orbits G(v) and G(e) are disjoint implies that forms a graph.Similar to the proof of Prop.2.1of [6] we can show that This completes the proof.Note.
is called the quotient graph induced by the fundamental domain (T; Y) for the action of G on X.

Quasi-graphs of groups Induced by Fundamental Domains
The concepts of quasi-graphs of groups and their fundamental groups introduced in [4] are modified and defined as follows.A quasi-graph of groups is defined to be a pair ) where Z is a connected quasi-graph and  is a mapping from Z into the class of all groups; where the image of each element (vertex or edge) Z x  under  is denoted by x  .i.e.
x x    ) ( such that for each edge eE(Z) the following hold. ( e e    ; (2) There exist monomorphisms denoted For each eE(Z), let e t be the value of e in Φ(Z; Γ;  ) where no confusion will be caused by the notations e t and t(e).It is clear that the relations of Φ(Z; Γ;  ) imply the following.(3) The fundamental group of Φ(Z, Γ) denoted Φ(Z; Γ) is defined to be the fundamental group Φ(Z; Γ;  ) relative to a maximal subtree  of Z.In view of above, Φ(Z; Γ) is independent of any maximal subtree of Z.The main result of this section is the following theorem.Theorem 3.1.Let G be a group acting on a connected quasi-graph X with inversions, and (Y; T) be a fundamental domain for the action of G on X.
It is easy to show that   This implies that the mapping ; Γ) forms a quasi-graph of groups.Let e be an edge of Y. Then Furthermore, the mapping : given by In the presentation (*), we  Proof.Let G be a fundamental group of a quasi-graph of groups Φ(Z, Γ) , where Z contains more than one vertex,  be a maximal subtree of Z, and H be a subgroup of G.We need to find a quasi-graph of groups ) ; ( , where H is its fundamental group.In [5], a tree X is constructed as follows.V(X) = {[g, v]: vV(Z), gG}, and E(X) = {[g, e]: eE(Z), gG}, where .The tree of representatives T for the action on of G on X is defined as V(T) = { [1, v]


. Then (T; Y) is a fundamental domain for the action of G on X.For each vertex vV(T), and each edge eE(Y), define the following.; Γ ).Consequently H is a quasi-graph group.This completes the proof.

( 1 )
Φ(Z; Γ;  ) is generated by elements e t and g, where g v forms a quasi-graph of groups, and the fundamental group T), and eE(Y).Furthermore, if X is a tree, then G quasi-graph.Now we show that Γ satisfies the conditions of the definition of quasi-graph of groups.If e is an edge of Y, then G(e) is an edge of )


. This leads the presentation of the fundamental group )

DEYY
is a double coset representative system for G mod (H, but otherwise arbitrary, gG.For more details of the structures of the action of H on X obtained in[7] is defined as follows.V( H T ) = {d(v): vV(T), d v D }, and E( H T ) is the set of edges,{ ab(e), ab( e ): e 0 consists of the edges of the forms ab(e), and ab( e )) can be formed as follows.0 E ( H Y ) = E( H T ) consists of the set of edges of the following forms.
of the set of edges of the following forms.
of the set of edges of the form: ab(x), where a ) Theorem 3.1 implies that H is the fundamental group of the quasi-graph of groups : vV(Z)}, and E(T ) = {[1, e]: eE(  )}.Also, [g, v]* = [1, v].The transversal Y for the action on of G on X consists of all edges