Abstract
PSL(2, R) is the most frequently studied subgroup of the Möbius transformations. By adding anti-automorphisms
Abstract
to the group PSL(2, R), the group G = PSL(2, R) ∪ G is obtained. The elements of this group correspond to matrices of GL(2, R). In this study, we consider the relationships between fixed points of the elements of the group G and eigenvectors of matrices corresponding to the elements of this group. MSC: Primary 20H10; 15A18
Introduction
Let C ∞ = C ∪ {∞} be the extended complex plane. A Möbius transformation is a function f of the form
Introduction
where a, b, c, d ∈ C and ad -bc = . Each Möbius transformation is a meromorphic bijection of C ∞ onto itself and is called an automorphism of C ∞ . Möbius transformations form a group with respect to composition. If T is a Möbius transformation, then the composition T • R is called an anti-automorphism of C ∞ , where R(z) = -z. The union of automorphisms and anti-automorphisms also form a group under the composition of functions.
Introduction
If coefficients of Möbius transformations are taken as real numbers, we obtain the most frequently studied subgroup of this group:
Introduction
By adding anti-automorphisms G = { a z+b c z+d
Introduction
The elements of this group correspond to the matrices of GL(, R). If we take T(z) ∈ G, then T(z) has the matrix presentation
Introduction
The fixed points of automorphisms and anti-automorphisms of the extended complex plane have especially been of great interest in many fields of mathematics, for example, in number theory, functional analysis, theory of complex functions, geometry and group theory (see [-] and references therein).
Introduction
In [], Beardon gave some relationships between the fixed points of Möbius maps and the lines of the eigenvectors of their corresponding matrices. So, these studies include the transformations of PSL(, R). In this study, we investigate similar relationships for transformations of G . Thus we complete the problem for the group G.
Preliminaries
In this section we give brief information about complex lines and fixed points of the transformations of G.
Definition  []
A complex line is a one-dimensional subspace of the vector space C ,t of
Definition  []
A complex line L is the set of complex scalar multiples of some non-zero point in C ,t , and so it is of the form
Definition  []
If z  = , we can form the quotients
Definition  []
of the coordinates of the non-zero points on the line L in Definition  and the common value of all of these quotients is the slope
Definition  []
The single complex line whose slope is not defined is
Definition  []
and, by convention, we say that this line has slope ∞. Given a complex number w, there is a unique complex line L(w) with slope w, namely
Theorem  [] Let f be a Möbius map with corresponding matrix A. Then f (w) = w if and only if L(w) is a line of eigenvectors of A.
Here we mention types of the elements in the group G briefly. For each T ∈ G, the point z ∈ C ∞ is called a fixed point of T if T(z) = z, and the trace of T(z) is defined by tr(T) = a+d. There is a relation between the fixed points and the trace of a transformation of G. Thus we can determine fixed points location in C ∞ with the trace.
Theorem  [] Let f be a Möbius map with corresponding matrix A. Then f (w) = w if and only if L(w) is a line of eigenvectors of A.
If T(z) ∈ PSL(, R), then the number of fixed points of T(z) is at most two. Also, if 
GL(, R). The characteristic polynomial for this matrix is
We use the eigenvector representation
GL(, R). The characteristic polynomial for this matrix is
for the matrix T. First we begin with the glide reflections.
Glide reflections
We will show that the fixed points of a glide reflection T(z) correspond to the two lines of eigenvectors for the matrix T corresponding to T(z). In the following two lemmas, we determine the eigenvalues and eigenvectors of the matrices which correspond to the glide reflections.
Lemma  Let the matrix T = a b c d correspond to any glide reflection. The eigenvalues of T are
and the eigenvectors of T are
Lemma  Let the matrix T = a b c d correspond to any glide reflection. The eigenvalues of T are
Proof It is easy to compute the eigenvalues by the condition a + d =  and (.). For an eigenvalue λ, we obtain the eigenvector by the following equation . Therefore we obtain the eigenvectors as
Theorem  Let T(z) be a glide reflection map in the group G with corresponding matrix T. Then T(w) = w if and only if L(w) is a line of eigenvectors of T.
Proof Let T(z) be a glide reflection map in the group G with a corresponding matrix T.
Theorem  Let T(z) be a glide reflection map in the group G with corresponding matrix T. Then T(w) = w if and only if L(w) is a line of eigenvectors of T.
For glide reflections, the lines with slope w, where w = a-d± 
Reflections
Recall that we have tr(T) =  for any reflection transformation. Proof By (.), if we use the condition tr(T) = , the result is obtained.
Lemma  Let the matrix T =
First we begin the case c = . For this case, the set of fixed points is a circle with radius ∞ (that is, a line on the complex plane).
Lemma  Let the matrix T = a b c d
correspond to any reflection with c = . We have
Lemma  Let the matrix T = a b c d
Proof Proof It is easy to compute the eigenvalues λ =  and λ = - by the condition a + d =  and (.). For these eigenvalues, we obtain the eigenvectors by the following equation
Lemma  Let the matrix T = a b c d
If we choose k  = r as a parameter, we find the eigenvectors as 
Proof
The proof follows by straightforward computations.
Proof
In the following theorem, we explain the relationship between fixed points of the reflections with c =  and eigenvectors of the matrices corresponding to those reflections. Finally, we consider the reflections with c = . Lemma  can be proven in a similar way as Lemma .

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