# SPHERICAL GROWTH SERIES OF THE FREE PRODUCT.

Byline: Zaffar Iqbal, Iftikhar Ahmad and Shamsa Aslam

ABSTRACTS

The computations of A. Machi for the spherical growth series of are given in [1]. We generalize this idea and compute the spherical growth series of the modular group.

Keywords: word length, free product, spherical growth series.

1. INTRODUCTION

In geometric group theory the spherical growth function is the study of combinatorial aspect of the elements in a finitely generated group . Spherical growth series provides an invariant of a group. A. Machi [1] compute the spherical growth series of are given. We generalize this idea and compute the spherical growth series of the modular group . We start with few basic definitions.

Definition 1. [1] Le be a finitely generated group and let and be a finite set of generators of and their inverses, respectively. The word length of is denoted by and is defined as the least non-negative integer n for which there exist such that .

Definition 2. [1] The spherical growth function of the group with generating set associates to a non-negative integer the number of the element such that , the number of elements of of word length . The spherical growth series (SGS) of is the formal power series

Equation

In this paper, for different groups with the given generating set , we consider the words of length exactly and find their corresponding spherical growth series.

Definition 3. [1] Let and be two groups. The free product of and is a group whose elements are the words of the form , where and with the condition that and are possibly identities of and , respectively.

The free product of two groups is a group that contains the elements of both and . In this product and are subgroups and the elements of these subgroups are generators of .

If and are non-trivial then is always infinite.

2. Spherical Growth Series of Free Groups and of Free Abelian Groups

There corresponds a growth function to each given group with a finite generating set . The growth function, an invariant of the group, generates a series of the group called the growth series of the group In this section we present SGS of the free groups and the free abelian groups. We start with a famous result regarding the SGS as a proposition.

Proposition 1 [3]. Let and be two groups generated by finite sets and , respectively. Then the SGS of the direct product , finitely generated by

Equations

3. Spherical Growth Series of

In this section we find the SGS of the free product . The computations of A. Machi for the SGS of are given in [1].

Example 6. [1] The SGS of the free product is given

Equations

In order to compute the SGS of the free product , we are using the following notations:

Equations

Table 1.

k###Set of words###a ( k ) n( X i )

1###X1###1

2###X2###n 1

###n 1

3###X3

4###X4###( n 1) 2

5###X5###( n 1) 2

6###X6###( n 1) 3

Also the sets of words of length ending at any word of 1 Y are given in the following table:

Table 2.

k###Set of words###b ( k ) n(Yi )

1###Y1###n 1

2###Y2###n 1

3###Y3###( n 1) 2

4###Y4###( n 1) 2

5###Y5###( n 1) 3

Now to find recurrence relations between the words of Table 1 and Table 2, we define a function P Q f : by . ) ( 1 wx w f It is clear that f is bijective. It follows that

Equations

4. Spherical Growth Series of Theorem.

The SGS of the free product is given by

Equations

Now the sets of words of length ending at any word of 1 X are written in the table given below.

Table 3.

k Set of words###a ( k ) n( X i )

1###X1###m 1

2###X2###(n 1)(m 1)

3###X3###(n 1)(m 1) 2

4###X4###(n 1) 2 (m 1) 2

5###X5###(n 1) 2 (m 1) 3

And the words of length ending at any word of 1Y are shown in the following table:

Table 4.

k###Set of words###b ( k ) n(Yi )

1###Y1###n 1

2###Y2###( n 1)(m 1)

###( m 1)(n 1) 2

3###Y3

4###Y4###( m 1) 2 ( n 1) 2

5###Y5###( m 1) 2 ( n 1) 3

To find recurrence relations between the words of Table 3 and Table 4, we define a bijective function

Equations

Unexpected End of Formula[1] Pierre de la Harpe, Topics in Geometric group theory", Chicago University Press, 2000.

[2] R. Grigorchuk, P. D. L. Harpe, On problems related to growth, entropy and spectrum in group theory", J DYN CONTROL SYSTJ., vol 3, no. 1, p. 51-89 (1997).

[3] Nick Scott, Growth of Finitely Generated Groups", Honors Thesis, The University of Melbourne, 2007.

ABSTRACTS

The computations of A. Machi for the spherical growth series of are given in [1]. We generalize this idea and compute the spherical growth series of the modular group.

Keywords: word length, free product, spherical growth series.

1. INTRODUCTION

In geometric group theory the spherical growth function is the study of combinatorial aspect of the elements in a finitely generated group . Spherical growth series provides an invariant of a group. A. Machi [1] compute the spherical growth series of are given. We generalize this idea and compute the spherical growth series of the modular group . We start with few basic definitions.

Definition 1. [1] Le be a finitely generated group and let and be a finite set of generators of and their inverses, respectively. The word length of is denoted by and is defined as the least non-negative integer n for which there exist such that .

Definition 2. [1] The spherical growth function of the group with generating set associates to a non-negative integer the number of the element such that , the number of elements of of word length . The spherical growth series (SGS) of is the formal power series

Equation

In this paper, for different groups with the given generating set , we consider the words of length exactly and find their corresponding spherical growth series.

Definition 3. [1] Let and be two groups. The free product of and is a group whose elements are the words of the form , where and with the condition that and are possibly identities of and , respectively.

The free product of two groups is a group that contains the elements of both and . In this product and are subgroups and the elements of these subgroups are generators of .

If and are non-trivial then is always infinite.

2. Spherical Growth Series of Free Groups and of Free Abelian Groups

There corresponds a growth function to each given group with a finite generating set . The growth function, an invariant of the group, generates a series of the group called the growth series of the group In this section we present SGS of the free groups and the free abelian groups. We start with a famous result regarding the SGS as a proposition.

Proposition 1 [3]. Let and be two groups generated by finite sets and , respectively. Then the SGS of the direct product , finitely generated by

Equations

3. Spherical Growth Series of

In this section we find the SGS of the free product . The computations of A. Machi for the SGS of are given in [1].

Example 6. [1] The SGS of the free product is given

Equations

In order to compute the SGS of the free product , we are using the following notations:

Equations

Table 1.

k###Set of words###a ( k ) n( X i )

1###X1###1

2###X2###n 1

###n 1

3###X3

4###X4###( n 1) 2

5###X5###( n 1) 2

6###X6###( n 1) 3

Also the sets of words of length ending at any word of 1 Y are given in the following table:

Table 2.

k###Set of words###b ( k ) n(Yi )

1###Y1###n 1

2###Y2###n 1

3###Y3###( n 1) 2

4###Y4###( n 1) 2

5###Y5###( n 1) 3

Now to find recurrence relations between the words of Table 1 and Table 2, we define a function P Q f : by . ) ( 1 wx w f It is clear that f is bijective. It follows that

Equations

4. Spherical Growth Series of Theorem.

The SGS of the free product is given by

Equations

Now the sets of words of length ending at any word of 1 X are written in the table given below.

Table 3.

k Set of words###a ( k ) n( X i )

1###X1###m 1

2###X2###(n 1)(m 1)

3###X3###(n 1)(m 1) 2

4###X4###(n 1) 2 (m 1) 2

5###X5###(n 1) 2 (m 1) 3

And the words of length ending at any word of 1Y are shown in the following table:

Table 4.

k###Set of words###b ( k ) n(Yi )

1###Y1###n 1

2###Y2###( n 1)(m 1)

###( m 1)(n 1) 2

3###Y3

4###Y4###( m 1) 2 ( n 1) 2

5###Y5###( m 1) 2 ( n 1) 3

To find recurrence relations between the words of Table 3 and Table 4, we define a bijective function

Equations

Unexpected End of Formula[1] Pierre de la Harpe, Topics in Geometric group theory", Chicago University Press, 2000.

[2] R. Grigorchuk, P. D. L. Harpe, On problems related to growth, entropy and spectrum in group theory", J DYN CONTROL SYSTJ., vol 3, no. 1, p. 51-89 (1997).

[3] Nick Scott, Growth of Finitely Generated Groups", Honors Thesis, The University of Melbourne, 2007.

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Publication: | Science International |
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Article Type: | Report |

Date: | Aug 31, 2015 |

Words: | 807 |

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