Automorphisms of certain relatively free groups and one relator groups

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AUTOMORPHISMS OF CE 豊 TAIN 豊電 LATI VELY FREE GROUPS AND ON 電 R 鴬 LATOR GROUPS

By

N. CHANDFIAMOWLISWARAN

Department of Mathematics

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fuif/ment of requirements for the award of the degree of DOCTOR OF PHILOSOPHY

INDIAN INSTITUTE OF TECHNOLOGY, DELHI

INDiA

NOVEMBER 1996

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Certi 丑 cate

-Fhis is to certify that tule thesis entitled "Automorphisrns of Certain Relatively Free Groups and One Relator Groups" whlich is being SUblflit,ted by N. Chandramowliswaran 1r the award of DOCTOR OF PHILOSOPHY (MATHEMATICS)to thle Indian Institute of Technology, Delhi, is a record of bonafide researchl 'vork carried ont by liiin unlder・mlly gui danlce ari (1 S tiper・vi s i oli.Tfie t}ic s is has reachec,l the standards fullfilling th（う reくluir・errients of the regulations relal・ing to t hc degree. TFie i'esiili・r; obtained in the i,1iesis hlave not beenl subiliitllecl to aity other Universit,y or I丁istitule 伽1・もhe award of i i.Ilう『 degree or diplollla.

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ACKN OWLED GEMENT S

I am greatly indebted to my supervisor Professor J.B. Srivastava for introducing me to Combinatorial Group Theory. His valuable guidance arid excellent teaching has beneffited me in building up the background for this research work. Bis tremendous patience and constant encouragement has been a source of motivation for me.

i take this opportunity to thank Professor S・R・K. lyengar, Head, Department of M戚hem航ics, for his encour昭ement and moral support in completing this thesis. The kind help and association extended by the faculty of Department of M誠，hem試ics, especially Professor N.S. Kambo and Professor B.R. Janda is gr_誠efully acknowledged. It is 叫pie闘ure to thank_町friends V. Ravindranath, T.V. Selva Kumaran, K. Srinivasaii,M.

Arulanandam, R.K. Dash, Necia Nataraj,Soma Gupta a,nd all others for keeping me in good spirits and cheers.

This research work is supported by the Council of Scientiffic and In- dustrial Research and I owe immense gr誠・itude to CSIR, for providing 丘nancial support.

I owe so much to my sisters Saradha, Viji，ぬtchala and my brother N.

罷nkatesan for their e_肥r loving wishes.

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N.

Chandramowliswaran

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Notations

C Proper inclusion

9 Inclusion

_i/v Set of natural numbers

2 Set of integers

H 5 G .g is a subgrcup of G

Nョ Y . N is a normal subgroup of C

z（の Center of G

ュ^y ^J_1xy

[x, y」ュー‘^{y-1x y}

[z1，ユ 2,… ，^xn^,xⁿ⁺¹^{j [[}xi，^エ^2,

…

^，エn，ュi ^{n+1 j}

争、^{( G)} The n th term of the lower centra,l series of G

島 ^(G) the (n+l)th term of the derived series of G [ H, K] The subgroup generated by all commutators

[h,k]。 h e H, k E K

G' The commutator subgroup of G j;; The free group on X:，及，…，ル

F The free group on Xi，義，…，瓦，，・・

ZC The integral Group ring of G

2凡 The integral Group ring of the free group P'?,

ハ凡 ^{The augm}帥^t肌ion ideal of Z FL

GL, (2凡） The general linear group of degree n over 2凡ム^R The left ideal ofZ凡 generated by all e)emcnts

of the form {r一 i s r E R, RS凡｝

OL,-. (R) The general linear group of degree n over the ring R OJJ。 ^{(2) rnI}、e group of n x n matrices with integra.l entries

whose determinant is 士」

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111111 IllX G G

JA(G)

End（の

響－奴四響＝改四

Automorphism group of G

The group of inner automorphisms of C The kernel of the natural homomorphism {Aut (G) →Aut (G/G')}

The semi-group of all endomorphisms of G The left Fox derivative of the丘eely reduced word W E凡 w.r.t乃

The right Fox derivative of the freely reduced word W E凡 w.r.t 毛

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ABSTRACT

As the title of the thesis "Autornorphisms of. Certain Relatively Free Groups arid Ciie Relator Croups" suggests, our main aim in this thesis is to attempt and solve some problems ori a.utomorphism groups of some relatively free groups and one relator groups of special type. In chapter 1,wehave discussed the lower centra,l series, the derived series, residually finite groups, varieties of groups and relatively free groups. i施have given a brief and up-to- date survey of known results relevant to our works in this thesis together with our comment.s and remarks. In chapter 2, we have obtained some results ori group rings and applied free di無rential calculus, developed by 恥X to get a new t田t word for determining cei・tahi automorphisms. We also discussed J.S. Birman's theorem and Krasnikov's theorem to get a relation between automorphisms arid invertible matrices over certain integral group rings. [n chapter 3,we h加obtained several important results ofthe thesis related to autoinorphisms of cci・tain relatively fr肥 groups. It contains several results which play a signifficant role in liれing automorphisrns and non・tame automorphisms of certain groups.

We hlave derived a very explicit criterion for determining whether certain special types of eridomorpiijsms of free groups of fluite rank are automorphisms. We have also constructed a class of non-tame automorpltisms of the groups F2/呪(R),mと3, where .8 is a non ti・i "ial normal subgroup of free group F2.

Chapter 4 deals with automorphisms of two generator one relator groups. For叫i'tai n two generator torsioii-free one relator group G, it has been shown that the autornorphism group Aut（のof G coincides with the group of IA-automorphisms of G. Several examples ai・e givenI・ln chapterり，we nave ootaincu a ciiaracしeri型凹四D L semi compiete groups anaiogrnユs

to Baer's directねctor property criterion for complete groups. Many more results and severa.l examples are given to demonstrate the situations where IA(G)=Jnrz(G). At the end we have givenl some concludling remarks and recorded 9 problems which arose during our work.

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74 i Introduction i.i Preliminaries

2 Automorphisms of free groups, free solvable groups and relatively free groups

i .3 A Brief Outline

2 Group Rings and Free Di卿rential Calculus

2.1 Group Rings ...

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2.2 Free Di

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rential Calculus

2.3 Automorphisms of Relatively Free Groups

3 Automorphisms of Some Relatively Free Groups 3.1 Automorphisrns of Certain Relatively Free Groups 3'2 Determination of Certain JA-Automorphisms

3,3 Some Applications 81

踊

4 One Relator Groups

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4.2 Automorphisms of Certain Torsion-free T如－genera切r

One Relator Groups . . . ．。． . . . .,. . ．、． . 90 4.3 Example . . . .. . . .,. . . .,. 99

5 Sorne More Results onAutomorphism Groups 105 5.1 Semicomplete Groups . . . .'. . . I 05 52 Some more Examples .,. . . .',. . . . 116 5.3 Concluding Remarks . . .,,．』‘ . . . .,. . i 20