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Article

Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group $▵*$(3,q,r) for q < r Primes

by
Sajida Younas
1,
Sajida Kousar
1,
Majed Albaity
2 and
Tahir Mahmood
1,*
1
Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan
2
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80348, Jeddah 22254, Saudi Arabia
*
Author to whom correspondence should be addressed.
Axioms 2023, 12(12), 1128; https://doi.org/10.3390/axioms12121128
Submission received: 11 October 2023 / Revised: 7 December 2023 / Accepted: 11 December 2023 / Published: 15 December 2023

Abstract

:
Hyperbolic triangle groups are found within the category of finitely generated groups. These are topological groups formed by the reflections along the sides of a hyperbolic triangle and acting properly discontinuously on the hyperbolic plane. Higman raised a question about the simplicity of finitely generated groups. The best known example of a simple group is the alternating group $A n$, where $n ≥ 5$. This article establishes a relation between the hyperbolic triangle group denoted as $▵ * ( 3 , 7 , r )$ and the alternating group. The approach involves employing coset diagrams to establish this connection. The construction of adjacency matrices for these coset diagrams is performed, followed by a detailed examination of their spectral characteristics.
MSC:
05E18; 65F15

1. Introduction

In the Euclidean plane for a line L and point p that is not on line L, there is exactly one line passing through p and parallel to L. The desire for more than one line passing through point p and not intersecting L laid down the basis for non-Euclidean planes—surfaces with unbounded straight lines. There are several models of the hyperbolic plane, but in this work, we consider only the unit disc $D 2 = { z ∈ C : | z | < 1 }$ with distance function $d s = | 2 d z | 1 − | z | 2$. The $D 2$-lines are the circular arcs perpendicular to the unit circle bounding $D 2$ or the diameters of $D 2$. The angle between two intersection lines is the same as defined in the case of Euclidean geometry. However, the sum of the internal angles of a hyperbolic triangle is less than $π$. The distance preserving maps—isometries—such as reflection, rotation, translation, etc., of any Euclidean plane or non-Euclidean plane are bijective functions and their composition is again an isometry. For instance, the composition of two reflections is a rotation [1,2]. Thus, the collection of isometries of $D 2$ forms a group under the composition of functions. If we consider a hyperbolic triangle, then the group generated by the reflections along the sides of the triangle is called a hyperbolic triangle group. The hyperbolic triangle groups are examples of finitely generated groups. Consider a hyperbolic triangle T with hyperbolic angles $π p$, $π q$ and $π r$ and vertices labeled by $A , B$ and C. Let $x 1 , x 2$ and $x 3$ be the reflections in the triangle’s sides $B C , A C$ and $A B$, respectively. Since the sum of the internal angles of a hyperbolic triangle is less than $π$, this gives:
$1 p + 1 q + 1 r < 1 .$
The hyperbolic triangle groups generated by the reflections $x 1 , x 2$ and $x 3$ are presented as:
$▵ * ( p , q , r ) = 〈 x 1 , x 2 , x 3 : x 1 2 = x 2 2 = x 3 2 = ( x 1 x 2 ) p = ( x 2 x 3 ) q = ( x 3 x 1 ) r = 1 〉 .$
Letting $x = x 1 x 2 , y = x 2 x 3$ and $t = x 2$, we have an alternate presentation for the hyperbolic triangle group $△ ★ ( p , q , r )$, which is,
$▵ * ( p , q , r ) = 〈 x , y , t : x p = y q = t 2 = ( x y ) r = ( x t ) 2 = ( y t ) 2 = 1 〉 .$
The collection of all bijective functions from a non-empty set X to itself forms a group under the composition of functions and is called a symmetric group. If X contains n elements, then $S n$ represents the symmetric group of X and has exactly $n !$ elements. The classical Cayley’s theorem, which states that every group can be embedded into a suitable symmetric group, makes these groups more attractive. The most important subgroup of $S n$ is the alternating group $A n$; $A n$ is the largest normal and simple (for $n ≥ 5$) subgroup of $S n$. At the start of the twentieth century, Miller [3] presented a remarkable result of combinatorial group theory. He proved that every alternating group except $A 6 , A 7$ and $A 8$ can be generated by two elements of order 2 and 3, and so these alternating groups are quotients of modular group $P S L ( 2 , Z ) = 〈 x , y : x 2 = y 3 = 1 〉$.
In 1968, Higman conjectured that every co-compact triangle group
$▵ ( p , q , r ) = 〈 x , y : x p = y q = ( x y ) r = 1 〉$
has among its homomorphic images, all but finitely, many alternating groups. Higman proved his claim for two triangle groups $▵ ( 2 , 3 , 7 )$ and $▵ ( 2 , 4 , 5 )$. Later, Conder [4] refined Higman’s method and proved the result for $▵ ( 2 , 3 , r )$, where $r ≥ 7$. Following their techniques, Mushtaq and Rota [5] examined $▵ ( 2 , k , l )$, for all even $k ≥ 6$. Mushtaq and Servatius [6] considered $▵ ( 2 , q , r )$ for almost all $5 ≤ q ≤ r$ and verified Higman’s conjecture. Everitt [7,8] set out results for $▵ ( p , q , r )$ where $p < q < r$ are primes by using combinatorial techniques. All these algebraists use coset diagrams to answer Higman’s conjecture. The triangle group $▵ ( p , q , r )$ considered by Everitt is generated by two elements with orders p and q, whereas the hyperbolic triangle group $▵ * ( p , q , r )$ is generated by three elements with orders p, q and 2. In the coset diagram of the hyperbolic triangle group, the vertical symmetry depicts the action of the generator of order 2. Thus, the coset diagrams of $▵ * ( p , q , r )$ should be vertically symmetric, unlike the coset diagram of $▵ ( p , q , r )$.
Coset diagrams are directed graphs (that is, digraphs), and over the years, matrices have been used to study various properties of graphs, including the joining of graphs, and to identify isomorphic images of graphs. For digraphs, numerous matrix representations are established such as adjacency, Laplacian and incidence. However, in spectral graph theory, the first matrices that were analyzed were adjacency matrices [9]. Patne and Avachar [10] introduced techniques for the composition of graphs and analyzed the disjunction operation in graphs. Harary [11] introduced formulas for the determination of adjacency matrices of ordinary graphs and digraphs. Elspas and Turner [12] proposed that when adjacency matrices have non-repeated eigenvalues and vertices of graph are prime numbers, then the condition of the isomorphism of two graphs will be met if and only if their connection sets are identical. The eigenvalues of a graph are precisely the eigenvalues of its adjacency matrix, whereas the sum of the absolute value of eigenvalues of a graph is known as the energy of the graph. Kaveh and Rahami [13] studied the procedure to obtain the eigenvalues and eigenvectors of adjacency and Laplacian matrices. Gutman and Zhou [14], Liu and Li [15], Das and Mojalal [16], Khan et al. [17], Cokilavany [18] and Ramezani [19] worked on the upper and lower bounds of the energy of a graph. Huang et al. [20] conducted a spectral analysis of the adjacency matrix of random geometric graphs. Litvak et al. [21] considered regular digraphs and discussed the structure of eigenvectors for them. The action of $P S L ( 2 , Z )$ on $P L ( F 5 n )$ yields a triangle group $▵ ( 2 , 3 , k ) = 〈 x , y : x 2 = y 3 = ( x y ) k = 1 〉$, where $k = 5$; that is, $A 5$. The coset diagrams and adjacency matrix for the action of $P S L ( 2 , Z )$ on $P L ( F 4 )$, $P L ( F 5 )$, $P L ( F 9 )$, $P L ( F 11 )$, $P L ( F 19 )$, $P L ( F 29 )$ and $P L ( F 59 )$ are described by Mushtaq and Rafiq [22]. The coset diagrams are directed graphs [23,24], so the relation between graphs and their adjacency matrices motivates us to understand the nature of adjacency matrices related to coset diagrams. In this study, by using the construction techniques of coset diagrams for the hyperbolic group $▵ * ( 33 , 7 . r )$ for all $r ≥ 11$, it is shown that these groups surject, all but finitely, many alternating groups. Additionally, this work examines the adjacency matrices, eigenvalues and energy of the coset diagram for $△ ★ ( 3 , 7 , r )$.

2. Preliminaries

Some fundamental definitions and ideas relevant to this work are presented in this section.

2.1. Group Action

For a group G and a non-empty set X, the action of G on X is precisely a map $· : X × G ⟶ X$ ($( x , g ) ↦ x · g$) such that for all $g 1 , g 2 ∈ G$ and $x ∈ X$, we have $x · ( g 1 g 2 ) = ( x · g 1 ) · g 2$ and $x · 1 = x$. The action of G on X gives rise to two important types of sets, namely, orbits and stabilizers [25,26]. The formation of orbits is based on the equivalence relation ∼ on X defined by $x ∼ y ⇔ y = x · g$ for some $g ∈ G$. The disjoint classes are called orbits of X under the action of G on X. If there is only one orbit under the action of G on X, then the action is said to be a transitive action. The collection of all orbits is called the orbit space of X by G and denoted by $X / G$. As $x · 1 = x$ for a fixed $x ∈ X$ it may be possible that some non-identity elements fix x, the collection of all $g ∈ G$ that fixes x (that is, $x · g = x$) is a subgroup of G, called the stabilizer of x in G and denoted by $G x .$
In this article, we deal with the group whose elements are bijective functions. The notion of the image of $x ∈ X$ under a function $f ∈ G$(group of bijective functions) provides a map
$· : X × G ⟶ X$
that is, for $f ∈ G$ and $x ∈ X$$( ( x , f ) ↦ ( x ) f ) .$ Thus, the composition of functions and images under the identity function implies that the map · defines an action of G on X. The orbit space under the action of a group of functions on a non-empty set is very important, especially when the groups of isometries of a plane are acting on that plane, as in this case where the orbit spaces turn into geometric surfaces [27].

2.2. Coset Diagrams

Let G be a group with presentation $〈 S | R 〉$, where $S = { s 1 , s 2 , … }$ is a set of generators and R is a set of relators in these generators. Suppose G acts on a set X. Then the elements of X can be represented by the vertices of a diagram, with an edge labeled by $s ∈ S$ directed from vertex $x i$ to vertex $x j$ exactly when s sends $x i$ to $x j$. Such an edge is called an s-arc. As G is acting on X, for all $s ∈ S$ and $x ∈ X$, there is a uniquely defined $x s ∈ X$ and also a unique $y ∈ X$ with $y s = x$. Thus, for both x and s, there is exactly one incoming and one outgoing s-arc at x. This includes the possibility of having a loop at x. If $r = s i 1 … s i k ∈ R$, then $r = 1$ in G. As $x 1 = x$, for all $x ∈ X$ we have that the path starting at x and following the edges labeled $s i 1 , s i 2 , …$ is closed. We call this path an r-face. Thus, the diagram has the following properties:
• For both $x ∈ X$ and generator s, there is exactly one incoming and one outgoing s-arc at the vertex x;
• For both x in X and $r = s i 1 … s i k ∈ R$, the path labeled $s i 1 , … , s i k$ starting at x is closed.
Conversely, consider $G = 〈 S | R 〉$ and a diagram D with vertex set X joined by s-arcs so that 1 and 2 above are satisfied. Define an action of G on X as follows: for $g ∈ G$ choose a word $g = s i 1 … s i k$ in the generators S, and for $x ∈ X$ let $x g$ be given by the following g-path that is a sequence of arcs labeled by $s i 1 , … , s i k$. To show this is well defined, consider $g = t i 1 … t i l$. Then $w = s i 1 … s i k t i l − 1 … t i 1 − 1$ is a word for the identity of G and so has to be a product of the conjugates of relator words. From condition 2, for $r ∈ R$ and any h, the r-path starting at $x h$ is closed, and so the $h r h − 1$-path starting at x is closed. This means that the w-path starting at x is also closed; hence, the $s i 1 , … , s i k$ and $t i 1 , … , t i l$ paths starting at x finish at the same vertex.
Choose the empty word for $1 ∈ G$ so that $x 1 = x$ for each $x ∈ X$. Let $g , g ′ ∈ G$ and let $g = s i 1 … s i k$ and $g ′ = s i 1 ′ … s i l ′$ be words for g and $g ′$. Then $s i 1 … s i k s i 1 ′ … s i l ′$ is a word for $g g ′$ so that $( x g ) g ′ = x ( g g ′ )$ for all $x ∈ X$. Thus, the diagram D gives an action of G on X.
The above discussion ensures that there is one-to-one correspondence between the actions of a group G on a set X, and diagrams whose vertices are the set X and edges are labeled by the generators of G such that 1 and 2 are satisfied. A diagram is said to be connected if any two points in the diagram are joined by a sequence of edges (taken in either direction). A connected diagram corresponds to a transitive action. The vertices of a diagram can be identified with the right cosets in G of the stabilizer $G x$ of a point x of X. For this reason, we call a diagram representing an action of G on a set X a coset diagram [28]. A Cayley diagram is a coset diagram where, in this correspondence, $G x$ is the trivial subgroup.
A diagram satisfying conditions 1 and 2 gives an action of G on X. Thus, for each $g ∈ G$ there is a map $σ g : X → X$ defined by $σ g : x ↦ x g$, and this is a permutation of X. The map $θ : G → S X$ defined by $θ : g ↦ σ g$ is a homomorphism, called the permutation representation of G corresponding to the diagram. Thus, each diagram on X satisfying conditions 1 and 2 corresponds to a homomorphism $G → S X$.
Consider the triangle group $Δ ( p , q , r )$ with $2 ≤ p < q$. The group $Δ ( p , q , r )$ is presented as:
$Δ ( p , q , r ) = 〈 x , y : x p = y q = ( x y ) r = 1 〉 .$
Let $θ : Δ ( p , q , r ) → S n$ be a homomorphism induced by an action of $Δ ( p , q , r )$ on a set V of n elements. For our convenience, the permutations $θ ( x ) , θ ( y )$ and $θ ( x y )$ are simply denoted by $x , y$ and $x y$. The coset diagram D for this action is defined by adopting the following conventions in addition to those of the previous section:
• q-cycles of y are depicted by shaded q-gons whose vertices are permuted in an anticlockwise direction, and fixed points of y by heavy dots;
• A p-cycle, say $( α 1 , … , α p )$, of x is represented by unlabeled arcs joining the vertices $α i$ to $α i + 1$ for $i = 1 … ( p − 1 )$, and $α p$ to $α 1$. The loops representing fixed points of x are omitted from the diagram.
In such a diagram D, the x-arcs and y-arcs are likely to overlap each other. To obtain a diagram that gives a clear picture of the actions of all the generators and relators, consider a 2-cell embedding of D into some surface. The embedding of a graph on a surface is a way to represent the graph on the surface such that the edges of the graph are represented by arcs that only intersect at their endpoints (see [29], p. 60). A 2-cell is a topological space that is homeomorphic to an open disc in the plane with the usual topology. A 2-cell embedding of a graph is an embedding with the property that the complement of the graph in the surface is a union of two cells. The 2-cells are called the faces of the embedding. The path—a sequence of arcs labeled by the generators—over the boundary of each face induces a path in the graph of the same length. It turns out that a 2-cell embedding of a graph can be determined by specifying at each vertex v of the graph a cyclic permutation $p v$ of the edges incident to v. Conversely, for a 2-cell embedding of a graph on a surface for each vertex, there is a corresponding ordering (that is, permutation) of the edges incident with that vertex. Thus, there is one-to-one correspondence between the 2-cell embeddings of a graph and the assignment of these permutations to its vertices ([29], Section 6-6).
In particular, for the hyperbolic triangle group, at each vertex of the diagram D, an arrangement of arcs corresponds to a 2-cell embedding on a closed orientable surface. Such an embedding of D on the surface gives a diagram with faces labeled by powers of x, powers of y or powers of $x y$, called x-faces, y-faces or $x y$-faces. In the embedded diagram, each $x y$-face corresponds to a cycle in the action of $x y$, and the length of that cycle can be found by counting the number of y-arcs adjacent to the face. It turns out that for all the diagrams in this article, the surface is the 2-sphere. By placing the 2-sphere with its south pole on a plane and using stereographic projection from the north pole, the diagram can be projected into the plane. If the diagram is arranged on the sphere in such a way that the north pole of the 2-sphere is in the center of an $x y$-face, then, after projection, this face will become the “outside” face of the diagram. It also turns out that all the planar diagrams in this work happen to have a “vertical line of symmetry”. This means that the action of $Δ ( p , q , r )$ depicted by a diagram can be extended to an action of $Δ * ( p , q , r )$ by defining the action of the generator t to be a reflection across this axis. That is, t is an involution. Due to the symmetrical arrangement of x-arcs and y-arcs along the vertical axis and their anticlockwise orientation, if x(y) sends a to b, then on the other side of vertical axis, x(y) $b t$ is sent to $a t$, and this gives $( x t ) 2 = 1$ ($( y t ) 2 = 1$). Thus, for $2 ≤ p < q$, any connected diagram with n vertices has the following properties:
• The q-cycles of y are represented by shaded q-gons permuted in an anticlockwise direction, and fixed points of y by heavy dots;
• A p-cycle, say $( α 1 , … , α p )$, of x is represented by arcs oriented in an anticlockwise direction, joining the vertices $α i$ to $α i + 1$ for $i = 1 … ( p − 1 )$, and $α p$ to $α 1$;
• The action of $x y$ satisfies $( x y ) r = 1$;
• The action of t is given by the reflection in a vertical line of symmetry.
Now, construct diagrams for $▵ * ( 3 , 7 , 11 )$ and boost them to obtain diagrams for $r ≥ 11$. The vertices of a heptagon not joined by any x-arc (that is, the fixed points of x) are called free vertices. These free vertices can be used to attach new y-arcs that will boost the diagram. Suppose there are $i = 1 , 2$ consecutive free vertices on a heptagon, say Q. By using x-arcs, one can join them with $3 − i$ free vertices of another heptagon, say $Q ′$. As the heptagons and x-arcs are both oriented in anticlockwise directions, there are $i − 1$ vertices of Q and $3 − i − 1 = 2 − i$ vertices of $Q ′$ fixed by $x y$. After adding a heptagon, there are $7 − ( i − 1 + 3 − i − 1 ) = 7 − 3 + 2 = 6$ new y-arcs in the face to which $Q ′$ was added. Thus, the length of the cycle of $x y$ corresponding to that face is increased by 6. If by using x-arcs the i consecutive free vertices on Q are joined to $3 − i$ fixed points of y, then the length of the cycle of $x y$ corresponding to that face will be increased by $k = 3 − i − ( i − 1 ) = 4 − 2 i$, called a type k-pendant. This gives that $3 − k + 1 2$ free vertices are required to attach a type k-pendant. As $1 ≤ i ≤ 3$, this gives $k = 0 , 2$. The type k-pendants are mostly added to the heptagons on the vertical line of symmetry.
The free vertices of the heptagon can be used to join diagrams, and as y-arcs give the cycle structure of $x y$ in the diagrams, x-arcs are preferred to join these diagrams. Two points, i and j, are fixed by x such that both t and $( x y ) α$ send i to j to form an $α$-handle and are denoted by $[ i , j ] α$. Suppose that $A 1 , A 2 , A 3$ are connected diagrams representing the transitive permutation representation of the group $▵ * ( 3 , 7 , r )$ where $r ≥ 11$. Let $n 1 , n 2 , n 3$ be the number of vertices used to construct these diagrams. If these diagrams contain $α$-handles $[ a 1 1 , a 2 1 ] α , ⋯ , [ a 1 3 , a 2 3 ] α$, then place them on a common vertical line of symmetry, adding 6 new x-arcs, joining the vertices of the handles by sending $a 1 i$ to $a 1 i + 1$ ($i = 1 , 2$) and $a 1 3$ to $a 1 1$, and $a 2 1$ to $a 2 3$ and $a 2 i$ to $a 2 i − 1$ for $i = 1 , 2$. Two new 3-cycles, $( a 1 1 , a 1 2 , a 1 3 )$ and $( a 2 1 , a 2 3 , a 2 2 )$, are added in the permutation representation of x.
As these two 3-cycles are permuted in anticlockwise directions along the vertical line of symmetry and in each handle t sends $a 1 i$ to $a 2 i$, then after this composition, $( x t ) 2 = 1$. Since only x-arcs join the diagrams, we have no change in the y-cycles and $y t$-cycles.
Suppose
$( a 1 1 , γ 1 1 , … , γ 1 α − 1 , a 2 1 , γ 1 α + 1 , … , γ 1 r − 1 )$
$( a 1 2 , γ 2 1 , … , γ 2 α − 1 , a 2 2 , γ 2 α + 1 , … , γ 2 r − 1 )$
$( a 1 3 , γ 3 1 , … , γ 3 α − 1 , a 2 3 , γ 3 α + 1 , … , γ 3 r − 1 )$
are the r-cycles of $x y$ passing through the handles. After composing the diagrams, these cycles become
$( a 1 1 , γ 2 1 , … , γ 2 α − 1 , a 2 2 , γ 1 α + 1 , … , γ 1 r − 1 )$
$( a 1 2 , γ 3 1 , … , γ 3 α − 1 , a 2 3 , γ 2 α + 1 , … , γ 2 r − 1 )$
$( a 1 3 , γ 1 1 , … , γ 1 α − 1 , a 2 1 , γ 3 α + 1 , … , γ 3 r − 1 )$
This gives that the lengths of the cycles passing through the handles is unchanged. The cycles of $x y$ not passing through the handles are not affected by joining the diagrams. Hence, in the diagram formed by joining the diagrams , $x 3 = y 7 = ( x y ) r = ( x t ) 2 = ( y t ) 2 = 1$. As the diagrams are connected, the diagram formed by joining them using x-arcs as stated above is also connected and represents a transitive permutation action of the group $▵ * ( 3 , 7 , r )$ where $r ≥ 11$ of degree $Σ i = 1 3 n i$. The diagram constructed by joining the diagrams is represented as . The above discussion is summarized with the following proposition.
Proposition 1.
If $A 1 , A 2 , A 3$ are diagrams representing transitive permutation representation of the group $▵ * ( 3 , 7 , r )$ where $r ≥ 11$, then $[ [ A 1 , A 2 , A 3 ] ]$ is also a diagram for $▵ * ( 3 , 7 , r )$ with $Σ i = 1 3 | A i |$ vertices.

3. Alternating Quotients of $▵ * ( 3 , 7 , r )$ Where $r ≥ 11$ Is Prime

To answer Higman’s claim, we use coset diagrams and achieve the following results.
Theorem 1.
(Jordan’s Theorem; [30], Theorem 13.9). Let G be a primitive permutation group of degree n containing a prime cycle for some prime $q ≤ n − 3$. Then G is either the alternating group $A n$ or the symmetric group $S n$.
The following standard lemma gives us criteria for the primitivity of an action. Here, by the support of a permutation $ν$ we mean the set of those elements that are not fixed by $ν$.
Lemma 1.
Let $G = 〈 x 1 , x 2 , … , x k 〉$ be a transitive permutation group of degree n containing a prime cycle ν. For each $x i$, suppose there is a point in the support of ν whose image under $x i$ is also in the support of ν. Then G is primitive.
A proof can be found in [8], Lemma 2.3.
To find the alternating quotients of $▵ * ( 3 , 7 , 11 )$, consider the diagrams $A ( 11 )$, $B ( 11 )$ and $C ( 11 )$ (see Figure 1 and Figure 2). They represent transitive permutation representation of the hyperbolic triangle group $▵ * ( 3 , 7 , 11 )$. Diagram A has two 1-handles, formed by the top two vertices of the upper two heptagons, and are denoted by $[ a 1 , a 2 ] 1$ and $[ a 1 ′ , a 2 ′ ] 1$. Similarly, the top two vertices of the upper two heptagons of B form two 1-handles and are denoted by $[ b 1 , b 2 ] 1$ and $[ b 1 ′ , b 2 ′ ] 1$. The top two vertices of the upper heptagon of C constitute a single 1-handle of C denoted by $[ c 1 , c 2 ] 1$. The handles $[ a 1 , a 2 ] 1$ and $[ a 1 ′ , a 2 ′ ] 1$, and $[ b 1 , b 2 ] 1$, $[ b 1 ′ , b 2 ′ ] 1$ and $[ c 1 , c 2 ] 1$ are used to join the diagrams. In these diagrams $| A ( 11 ) | = 33$ and $| B ( 11 ) | = 34$ are relatively prime. Now by adding heptagons and type 2-pendants to these diagrams, we can find diagrams for all $r ≥ 11$. If in the diagrams $A ( 11 )$, $B ( 11 )$ and $C ( 11 )$ a chain of heptagons is attached to the boosting points indicated by black squares, then each heptagon will add 6 new y-arcs to the face containing it. Thus, if there are heptagons $h ( h ≥ 1 )$, this gives diagrams for $r h = r 0 + 6 h$ where $r 0 = 11$. Two odd numbers $r h + 2$ and $r h + 4$ are left between $r h$ and $r h + 1$, and for these numbers, attach type 2-pendants with the last copy of the heptagon in the chains. To find the diagrams for $r 0 + 2 = 13$, attach type 2-endants to the boosting points in the diagrams. Note that while boosting these diagrams, equal numbers of vertices are added to each face of the diagrams $A ( 11 )$ and $B ( 11 )$. This means that for all $r ≥ 11$, the diagrams $A ( r )$ and $B ( r )$ have a relatively prime number of vertices. The cycle structure of $x − 1 y$ in these diagrams is given in Table 1.
For $r ≥ 11$, the diagram $C ( 11 )$ has a 23-cycle (labeled by the small circles in the diagram $C ( 11 )$) with no other cycle having a length divisible by 23, and this 23-cycle contains fixed points of x and t, an element and its image under y. Thus, the action is primitive for $q = 7$ and $r ≥ 11$, and the group $▵ * ( 3 , 7 , r )$ surjects either $A n$ or $S n$ for, all but finitely, many n.
For some values of r the generator t acts as an odd permutation in the diagrams $A ( r )$ and $B ( r )$. If it is odd in the resultant diagram, we construct another diagram $D ( 11 )$ (see Figure 2). A copy of the diagram $D ( r )$ does not affect the primitivity of the action as the cycle structures of $x − 1 y$ in $D ( r )$ are:
$D ( 11 ) : ( d 1 d 2 11 ) 6 2 D ( 13 ) : ( d 1 d 2 13 ) 6 8$
As all the generators act as even permutations, we obtain the resultant group as $A n$. This means that for $r ≥ 11$, the group $▵ * ( 3 , 7 , r )$ surjects almost all alternating groups. Hence, the result holds for the hyperbolic triangle group $▵ * ( 3 , 7 , r )$, where $r ≥ 11$ is a prime.

4. Adjacency Matrices for the Coset Diagrams of $▵ * ( 3 , 7 , r )$ Where $r ≥ 11$

This section provides an analysis of the adjacency matrices for the coset diagrams A, B and C constructed in the previous section. For a graph G with vertex set V, the adjacency matrix of G is square M consisting of two types of entries 0 and 1. The vertices of G are enlisted in rows and columns outside the matrix; that is, they are not a part of the adjacency matrix but are used to indicate the placement of 0 and 1. For instance, if the vertex i is connected to the vertex j by an edge, then it corresponds to 1 at the $( i j )$th place of the matrix, otherwise it is 0. In the case of directed graphs, we follow the direction of the edge; for instance, if i is connected to the vertex j by an edge, then it corresponds to 1 at the $( i j )$th place and 0 at the $( j i )$th place of the matrix. However, in the case of an undirected graph, we obtain 1 at both the $( i j )$th and $( j i )$th places of the matrix.
For the coset diagram E of $▵ * ( 3 , 7 , 11 )$ in Figure 3, the anticlockwise direction of the action of x and y on the set ${ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 }$ constitutes the following adjacency matrix.
The adjacency matrices for directed graphs are not symmetric.
The boosting and joining of diagrams $A ( 11 ) , B ( 11 )$ and $C ( 11 )$ provide us with the alternating quotients of $▵ * ( 3 , 7 , r )$ where $r ≥ 11$. After joining these three coset diagrams, we gain a connected graph $D 1$; that is, $D 1 = A ( 11 ) + B ( 11 ) + C ( 11 ) .$ The diagram $D 1$ has $33 + 34 + 50 = 117$ vertices and represents a transitive permutation representation of $▵ * ( 3 , 7 , 11 ) .$ The order of the adjacency matrix of $D 1$ is the $117 × 117$ matrix with $( i , j ) t h$ entry equal to 1 if vertices i and j are adjacent; that is, they are connected by an x or y edge in an anticlockwise direction and are otherwise equal to 0.
As in the previous section, we proved that $▵ * ( 3 , 7 , 11 )$ surjects the alternating group $A 117$. To obtain the alternating groups of greater order as a homomorphic image of $▵ * ( 3 , 7 , 11 )$, we use the handles $A H 1 = [ a 1 , a 2 ] 1$, $A H 2 = [ a 1 ′ , a 2 ′ ]$, $B H 1 = [ b 1 , b 2 ] 1$ and $B H 2 = [ b 1 ′ , b 2 ′ ]$ of $A ( 11 )$ and $B ( 11 )$ to join more copies of $A ( 11 )$ and $B ( 11 )$. Suppose $A H 1 , B H 1$ and $C H 1$ are used to construct $D 1$, the handle $A H 2$ of A is free so we can attach two copies, $A 1 ( 11 )$ and $A 2 ( 11 )$, with A by using $A H 2$, $A 1 H 2$ and $A 2 H 2$. The same can be carried out with $B ( 11 )$. The diagram $D 2 = A ( 11 ) + A 1 ( 11 ) + A 2 ( 11 ) + B ( 11 ) + B 1 ( 11 ) + B 2 ( 11 ) + C ( 11 )$ has $3 ( 33 ) + 3 ( 34 ) + 50 = 251$ vertices and depicts that $▵ * ( 3 , 7 , 11 )$ surjects the alternating group $A 251$. The handles $A 1 H 1$, $A 2 H 1$, $B 1 H 1$ and $B 2 H 1$ can be used to join one more copy of $A ( 11 )$ and $B ( 11 )$. Then the resultant diagram $D 3$ has 318 vertices. To keep the order of A and B relatively prime we must attach an equal number of copies of A and copies of B. If we attach m copies of A and B, then we will obtain diagram
$D m = D m − 1 + 67 ( 2 ) f o r m = 3 , 5 , 7 , ⋯$
and
$D m = D m − 1 + 67 ( 1 ) f o r m = 2 , 4 , 6 , ⋯$
The alternate utilization of handles corresponds to the increase in the order of alternating groups. Essentially, we explore the outcomes of combining multiple copies of A and B with different handles, resulting in distinct adjacency matrices.
The joining of diagrams provides us with alternating quotients of the group $▵ * ( 3 , 7 , 11 )$. However, boosting by heptagon and type 2-pendants results int the alternating quotients of the group $▵ * ( 3 , 7 , r )$, where $r > 11$ is prime. If we boost $A ( 11 )$, $B ( 11 )$ and $C ( 11 )$ by n heptagons, then each heptagon adds 6 new y-arcs in each face and the diagrams represent the action of $▵ * ( 3 , 7 , 11 + 6 k )$. Thus, the resultant diagram $D B k H = A ( 11 + 6 k ) + B ( 11 + 6 k ) + C ( 11 + 6 k )$ has $| A ( 11 + 6 k ) | + | B ( 11 + 6 k ) | + | C ( 11 + 6 k ) | = ( 33 + 3 ( 7 k ) ) + ( 34 + 3 ( 7 k ) ) + ( 50 + 4 ( 7 k ) ) = 117 + 70 k$ vertices. If we boost the diagrams by type 2-pendants, then each corresponds to two new y-arcs in each face, so the diagrams represent the transitive action of $▵ * ( 3 , 7 , 11 + 2 )$. Thus, the resultant diagram $D B P = A ( 13 ) + B ( 13 ) + C ( 13 )$ has 137 vertices. If type 2-pendants are attached with m copies of A and B, then the following diagrams are obtained.
$( D B P ) m = ( D B P ) m − 1 + 79 ( 2 ) f o r m = 2 , 4 , 6 , ⋯$
and
$( D B P ) m = ( D B P ) m − 1 + 79 ( 1 ) f o r m = 1 , 3 , 5 , ⋯$
Boosting by k heptagons and type 2-pendants represents the transitive action of $▵ * ( 3 , 7 , 13 + 6 k )$. Thus, the resultant diagram $D B k H P = A ( 13 + 6 k ) + B ( 13 + 6 k ) + C ( 13 + 6 k )$ has $137 + 70 k$ vertices.
The joining of boosted diagrams A, B and C gives diagrams $( D B n H ) m$, $( D B P ) m$ and $( D B k H P ) m$ with vertices of $117 + 70 k + ( 67 + 42 k ) m$, $137 + 79 m$ and $137 + 70 k + ( 79 + 42 k ) m$, respectively. We summarize the above discussion in the form of the following results.
Theorem 2.
The hyperbolic triangle group $▵ * ( 3 , 7 , 11 )$ surjects onto the alternating group $A n$, where $n ≥ 117 + 67 m ( m ≥ 0 a n d m ∉ 3 k − 2 ∣ k ∈ N )$. The action of $▵ * ( 3 , 7 , 11 )$ on $n ≥ 117 + 67 m ( m ≥ 0 a n d m ∉ 3 k − 2 ∣ k ∈ N )$ constitutes adjacency matrix $M D m$ having dimensions of $( 117 + 67 m ) × ( 117 + 67 m )$ and
(1)
Trace$( M D 2 n ) = T r a c e ( M D 2 n − 1 ) + 70$
(2)
Trace$( M D 2 n + 1 ) = T r a c e ( M D 2 n ) + 29$ for all n
Proof.
(1)
Consider the adjacency matrices $M D 2 n − 1$ of coset diagrams $D 2 n − 1$ resulting from the attachment of $A ( 11 )$, $B ( 11 )$ and $C ( 11 )$ using handles $H 1$. Let $d ´ 1 , d ´ 2 , … , d ´ 2 n − 1$ denote the diagonal elements of $M D 2 n − 1$.
Then the trace of $M D 2 n − 1$ where only one handle is used can be expressed as:
$Tr ( M D 2 n − 1 ) = ∑ p = 1 2 n − 1 d ´ p$
To obtain $D 2 n$, attach two copies of $A ( 11 )$ and $B ( 11 )$ by handle $H 2$ with $D 2 n − 1$. This process consumes four free vertices from $A ( 11 )$ and $B ( 11 )$, of $D 2 n − 1$. Additionally, two more copies of $A ( 11 )$ and $B ( 11 )$ are attached with 38 and 36 free vertices, respectively. Consequently, the total number of free vertices in $D 2 n$ is
$∑ p = 1 2 n − 1 d ´ p − 4 + 38 + 36 .$
Let $M D 2 n$ represent adjacency matrices of $D 2 n$ with diagonal elements $d 1 , d 2 , … , d 2 n$.
Thus, the trace of adjacency matrices $M D 2 n$ can be calculated as:
$Tr ( M D 2 n ) = Tr ( M D 2 n − 1 ) − 4 + 74 = Tr ( M D 2 n − 1 ) + 70$
(2)
Let $D 2 n$ be a diagram with an adjacency matrix $M D 2 n$. By appending an additional single copy of $A ( 11 )$ and $B ( 11 )$ to the remaining handles $H 1$ of $D 2 n$, we obtain $D 2 n + 1$ with its corresponding adjacency matrix denoted as $M D 2 n + 1$.
In this process, eight free vertices are used from copies of $A ( 11 )$ and $B ( 11 )$ in $D 2 n$, which reduces the total number of free vertices to
$∑ q = 1 2 n d q − 8 .$
Simultaneously, we attach one more copy of $A ( 11 )$ and $B ( 11 )$ with 19 and 18 free vertices, respectively. Consequently, the total number of free vertices in
$M D 2 n + 1 = ∑ q = 1 2 n d q − 8 + 19 + 18 = ∑ q = 1 2 n d q + 29 .$
Hence, the trace of $M D 2 n + 1$ can be calculated as:
$Tr ( M D 2 n + 1 ) = Tr ( M D 2 n ) − 8 + 37 = Tr ( M D 2 n ) + 29 .$
The spectral analysis of some of these matrices is explained in Table 2.
The reliability of the subsequent findings were assured by using the same procedures and frameworks as in Theorem 2.
Theorem 3.
The hyperbolic triangle group $▵ * ( 3 , 7 , 13 )$ surjects the alternating group $A n$, where $n ≥ 137 + 79 m ( m ≥ 0 )$. The action of $▵ * ( 3 , 7 , 13 )$ on $n ≥ 137 + 79 m ( m ≥ 2 )$ constitutes $( 137 + 79 m )$-dimensional adjacency matrix $M ( D B P ) m$ with
$T r a c e ( M ( D 2 n B P ) m ) = T r a c e ( M ( D 2 n − 1 B P ) m ) + 84$
$T r a c e ( M ( D 2 n + 1 B P ) m ) = T r a c e ( M ( D 2 n B P ) m ) + 35$
for all $n = 1 , 2 , …$
The spectral analysis of some of these matrices is explained in Table 3.
Theorem 4.
The hyperbolic triangle group $▵ * ( 3 , 7 , 11 + 6 k )$, where $k ≥ 2$ surjects onto the alternating group $A n$, and where $n ≥ 117 + 70 k + ( 67 + 42 k ) m$. The action of $▵ * ( 3 , 7 , 11 + 6 k )$ on $n ≥ 117 + 70 k + ( 67 + 42 k ) m$ constitutes $( 117 + 70 k + ( 67 + 42 k ) m )$-dimensional adjacency matrix $M ( D B k H ) m$ with
$T r a c e ( M D n ) = T r a c e ( M D n − 1 ) + 40 ∀ n = 1 , 2 , . . .$
The spectral analysis of some of these matrices is explained in Table 4.
Theorem 5.
The hyperbolic triangle group $▵ * ( 3 , 7 , 13 + 6 k )$, where $k ≥ 2$ surjects onto the alternating group $A n$, and where $n ≥ 137 + 70 k + ( 79 + 42 k ) m$. The action of $▵ * ( 3 , 7 , 13 + 6 k )$ on $n ≥ 137 + 70 k + ( 79 + 42 k ) m$ constitutes $137 + 70 k + ( 79 + 42 k ) m$-dimensional adjacency matrix $M ( D B k H ) m$ with
$T r a c e ( M D n ) = T r a c e ( M D n − 1 ) + 40 ∀ n = 1 , 2 , . . .$
The spectral analysis of these matrices is explained in Table 5.

5. Conclusions

This study examined the hyperbolic triangle groups $▵ * ( 3 , 7 , r )$ and proved that they have, finitely, many alternating groups among their homomorphic images. The action of $▵ * ( 3 , 7 , 11 )$ on $n ≥ 117 + 67 m ( m ≥ 0 )$ constitutes the adjacency matrix $M D m$ with dimensions of $( 117 + 67 m ) × ( 117 + 67 m )$ and Trace$( M D 2 n ) = T r a c e ( M D 2 n − 1 ) + 70$ and Trace$( M D 2 n + 1 ) = T r a c e ( M D 2 n ) + 29 .$ The action of $▵ * ( 3 , 7 , 13 )$ on $n ≥ 137 + 79 m ( m ≥ 0 )$ constitutes the $( 137 + 79 m )$-dimensional adjacency matrix $M D B P m$ with $T r a c e ( M ( D 2 n B P ) m ) = T r a c e ( M ( D 2 n − 1 B P ) m ) + 84$ and $T r a c e ( M ( D 2 n + 1 B P ) m ) = T r a c e ( M ( D 2 n B P ) m ) + 35$. The action of $▵ * ( 3 , 7 , 13 + 6 k )$ on $n ≥ 137 + 70 k + ( 79 + 42 k ) m$ constitutes the $137 + 70 k + ( 79 + 42 k )$ m-dimensional adjacency matrix $M ( D B k H ) m$ with $T r a c e ( M ( D n ) = T r a c e ( M ( D ( n − 1 ) ) ) + 40 ∀ n = 1 , 2 , ⋯ n .$ The adjacency matrices for the coset diagrams of these groups are non-invertible antisymmetric matrices with non-zero energy greater than $127.7403$.

Author Contributions

Conceptualization, S.Y. and T.M.; methodology, S.K.; validation, S.Y., S.K. and M.A.; formal analysis, T.M.; investigation, S.K.; writing—original draft preparation, S.Y.; writing—review and editing, S.K.; visualization, T.M.; supervision, M.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

No data are used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Coset diagrams $A ( 11 )$ and $B ( 11 )$ for $▵ * ( 3 , 7 , 11 )$.
Figure 1. Coset diagrams $A ( 11 )$ and $B ( 11 )$ for $▵ * ( 3 , 7 , 11 )$.
Figure 2. Coset diagrams $C ( 11 )$ and $D ( 11 )$ for $▵ * ( 3 , 7 , 11 )$.
Figure 2. Coset diagrams $C ( 11 )$ and $D ( 11 )$ for $▵ * ( 3 , 7 , 11 )$.
Figure 3. Coset diagram E for $▵ * ( 3 , 7 , 11 )$.
Figure 3. Coset diagram E for $▵ * ( 3 , 7 , 11 )$.
Table 1. Cycle structure of $x − 1 y$ in A, B and C.
Table 1. Cycle structure of $x − 1 y$ in A, B and C.
rDiagramVerticesCycle Structure of $x − 1 y$
11A33$( a 1 a 2 9 ) ( a 1 ′ a 2 ′ 9 ) 11$
B34$( b 1 b 2 9 ) ( b 1 ′ b 2 ′ 9 ) 2 10$
C50$( c 1 c 2 7 ) 2 5 2 6 23$
13A39$( a 1 a 2 11 ) ( a 1 ′ a 2 ′ 11 ) 13$
B40$( b 1 b 2 11 ) ( b 1 ′ b 2 ′ 11 ) 2 12$
C58$( c 1 c 2 9 ) 2 7 2 8 23$
$11 + 6 h$A$33 + 21 h$$( a 1 a 2 10 ) ( a 1 ′ a 2 ′ 10 ) 6 3 7 3 ( h − 1 ) 12$
B$34 + 21 h$$( b 1 b 2 10 ) ( b 1 ′ b 2 ′ 10 ) 2 6 3 7 3 ( h − 1 ) 11$
C$50 + 28 h$$( c 1 c 2 8 ) 2 6 6 7 1 + 4 ( h − 1 ) 23$
$13 + 6 h$A$39 + 21 h$$( a 1 a 2 10 ) ( a 1 ′ a 2 ′ 10 ) 8 3 7 3 ( h − 1 ) 12$
B$40 + 21 h$$( b 1 b 2 10 ) ( b 1 ′ b 2 ′ 10 ) 2 8 3 7 3 ( h − 1 ) 11$
C$58 + 28 h$$( c 1 c 2 8 ) 2 6 2 7 1 + 4 ( h − 1 ) 8 4 23$
$15 + 6 h$A$41 + 21 h$$( a 1 a 2 10 ) ( a 1 ′ a 2 ′ 10 ) 7 3 ( h − 1 ) 10 3 12$
B$42 + 21 h$$( b 1 b 2 10 ) ( b 1 ′ b 2 ′ 10 ) 2 7 3 ( h − 1 ) 10 3 11$
C$60 + 21 h$$( c 1 c 2 8 ) 2 6 2 7 1 + 4 ( h − 1 ) 10 4 23$
Table 2. Spectral analysis of $M D m$.
Table 2. Spectral analysis of $M D m$.
$M D 1$117062$127.7403$
$M D 2$2510132$272.4878$
$M D 3$3180161$341.9508$
$M D 4$4520231$493.3243$
Table 3. Spectral analysis of $M ( D B P ) m$.
Table 3. Spectral analysis of $M ( D B P ) m$.
$M ( D 1 B P ) 0$137072149.7791
$M ( D 2 B P ) 2$2950154324.8379
$M ( D 3 B P ) 3$3740189409.2571
Table 4. Spectral analysis of $M ( D B k H ) m$.
Table 4. Spectral analysis of $M ( D B k H ) m$.
$M ( D B 1 H ) 0$1870102199.3871
$M ( D B 2 H ) 0$2570142286.1425
$M ( D B 3 H ) 0$3270182365.0140
Table 5. Spectral analysis of $M ( D B k H P ) m$.
Table 5. Spectral analysis of $M ( D B k H P ) m$.
$M ( D B 2 H P ) 0$2770112214.6266
$M ( D B 3 H P ) 0$3470158299.3381
$M ( D B 4 H P ) 0$4170192378.7628
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Younas, S.; Kousar, S.; Albaity, M.; Mahmood, T. Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group $▵*$(3,q,r) for q < r Primes. Axioms 2023, 12, 1128. https://doi.org/10.3390/axioms12121128

AMA Style

Younas S, Kousar S, Albaity M, Mahmood T. Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group $▵*$(3,q,r) for q < r Primes. Axioms. 2023; 12(12):1128. https://doi.org/10.3390/axioms12121128

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Younas, Sajida, Sajida Kousar, Majed Albaity, and Tahir Mahmood. 2023. "Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group $▵*$(3,q,r) for q < r Primes" Axioms 12, no. 12: 1128. https://doi.org/10.3390/axioms12121128

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