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Chapter 1.
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Chapter 2.
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[15]
Chapter 3.

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[]\OT1/ptm/m/n/10.95 More gen-eral ma-chines are ob-tained if one al-lows the s
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[]\OT1/ptm/m/n/10.95 The maps from the state-set of [] and [] to the state-set 
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[]\OT1/ptm/m/n/10.95 If a ma-chine [] was cre-ated as a min-i-mized group/monoi
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[]\OT1/ptm/m/n/10.95 If [] was cre-ated us-ing \OT1/pcr/m/n/10.95 AsGroupFRMach
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Chapter 4.
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[]\OT1/ptm/m/n/10.95 This func-tion con-structs a new FR el-e-ment. [] is a lis
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[32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
Chapter 5.
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[][]\OT1/ptm/b/n/14.4 Operations and At-tributes for \OT1/pcr/b/n/14.4 MealyMac
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[]\OT1/ptm/m/n/10.95 The sec-ond com-mand is a short-hand for \OT1/pcr/m/n/10.9
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[]\OT1/ptm/m/n/10.95 This com-mand cre-ates a new Mealy ma-chine \OT1/pcr/m/n/1
0.95 n\OT1/ptm/m/n/10.95 , with state-set the nu-cleus (see
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[]\OT1/ptm/m/n/10.95 This nu-cleus ma-chine is char-ac-ter-ized as the small-es
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[56] [57]
Chapter 6.

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[][]\OT1/ptm/b/n/14.4 Methods and op-er-a-tions for \OT1/pcr/b/n/14.4 LinearFRM
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[]\OT1/ptm/m/n/10.95 If [] is a Mealy ma-chine/element, the re-sult is a vec-to
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Chapter 7.

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[]\OT1/ptm/m/n/10.95 The max-i-mal length of words to con-sider in the search i
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he func-tion, as
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\OT1/pcr/m/n/10.95 FRGroupByVirtualEndomorphism(f:MealyElement)\OT1/ptm/m/n/10.
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[]\OT1/ptm/m/n/10.95 This func-tion re-turns \OT1/pcr/m/n/10.95 true \OT1/ptm/m
/n/10.95 if all el-e-ments of [] have the \OT1/ptm/m/it/10.95 open set con-di-t
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[]\OT1/ptm/m/n/10.95 This is the small-est group/monoid/semigroup \OT1/pcr/m/n/
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\OT1/pcr/m/n/10.95 FullSCMonoid(AlphabetOfFRSemigroup(g),P) \OT1/ptm/m/n/10.95 
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\OT1/ptm/m/n/10.95 ([][]10.1.2[][]), \OT1/pcr/m/n/10.95 GrigorchukGroup \OT1/pt
m/m/n/10.95 ([][]10.1.9[][]), \OT1/pcr/m/n/10.95 GrigorchukOverGroup \OT1/ptm/m
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Chapter 8.

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[]\OT1/ptm/m/n/10.95 Each [] is of the form \OT1/pcr/m/n/10.95 "name=[[...],...
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\OT1/pcr/m/n/10.95 "name=[[...],...,[...]]:out"\OT1/ptm/m/n/10.95 , namely a ma
-trix whose en-tries are al-ge-braic ex-pres-sions
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Chapter 9.
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Chapter 10.

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[]\OT1/ptm/m/n/10.95 These are the fini-tary, bounded, polynomial-growth, finit
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\OT1/ptm/m/n/10.95 act-ing on the bi-nary tree. They are re-spec-tively short-c
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\OT1/pcr/m/n/10.95 FullSCGroup([1..2],l)\OT1/ptm/m/n/10.95 , \OT1/pcr/m/n/10.95
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\OT1/pcr/m/n/10.95 FullSCGroup([1..2],IsBoundedFRSemigroup)\OT1/ptm/m/n/10.95 ,
 \OT1/pcr/m/n/10.95 FullSCGroup([1..2],IsPolynomialGrowthFRSemigroup)\OT1/ptm/m
/n/10.95 ,
 []

[100

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[]\OT1/ptm/m/n/10.95 As par-tic-u-lar ex-am-ples, \OT1/pcr/m/n/10.95 BinaryKnea
dingMachine() \OT1/ptm/m/n/10.95 is the adding ma-chine;
 []


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\OT1/pcr/m/n/10.95 BinaryKneadingGroup() \OT1/ptm/m/n/10.95 is the adding ma-ch
ine; and \OT1/pcr/m/n/10.95 BinaryKneadingGroup("1") \OT1/ptm/m/n/10.95 is
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[101]
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[]\OT1/ptm/m/n/10.95 These are re-spec-tively the same as \OT1/pcr/m/n/10.95 Ad
dingGroup(2)\OT1/ptm/m/n/10.95 , \OT1/pcr/m/n/10.95 AddingMachine(2) \OT1/ptm/m
/n/10.95 and
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[]\OT1/ptm/m/n/10.95 The first func-tion con-structs the group gen-er-ated by t
he mixer ma-chine. For ex-am-ples
 []

[102]
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[]\OT1/ptm/m/n/10.95 These groups are \OT1/pcr/m/n/10.95 MixerGroup \OT1/ptm/m/
n/10.95 ([][]10.1.6[][])s. The most fa-mous ex-am-ple is
 []


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[]\OT1/ptm/m/n/10.95 These groups are all 4-generated and in-fi-nite. They are 
de-scribed in [[]][].
 []


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\OT1/pcr/m/n/10.95 GrigorchukGroups([1]) \OT1/ptm/m/n/10.95 is in-fi-nite di-he
-dral. If [] con-tains at least 2 dif-fer-ent dig-
 []

[103]
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[]\OT1/ptm/m/n/10.95 This is Grig-orchuk's first group, in-tro-duced in [[]][].
 It is a 4-generated in-fi-
 []


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\OT1/ptm/m/n/10.95 nite tor-sion group, and has in-ter-me-di-ate word growth. I
t could have been de-fined as
 []


Underfull \hbox (badness 10000) in paragraph at lines 6477--6479
[]\OT1/ptm/m/n/10.95 The com-mand \OT1/pcr/m/n/10.95 EpimorphismFromFpGroup(Gri
gorchukGroup,n) \OT1/ptm/m/n/10.95 con-structs an ap-
 []


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\OT1/ptm/m/n/10.95 prox-i-mat-ing pre-sen-ta-tion for the Grig-orchuk group, as
 proven in [[]][]. Adding
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\OT1/ptm/m/n/10.95 the re-la-tions \OT1/pcr/m/n/10.95 Image(sigma^(n-2),(a*d)^2
)\OT1/ptm/m/n/10.95 , \OT1/pcr/m/n/10.95 Image(sigma^(n-1),(a*b)^2) \OT1/ptm/m/
n/10.95 and
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[]\OT1/ptm/m/n/10.95 This is a group with same clo-sure as the Grig-orchuk grou
p (see
 []


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\OT1/pcr/m/n/10.95 GrigorchukGroup \OT1/ptm/m/n/10.95 ([][]10.1.9[][])), but no
t iso-mor-phic to it. It could have been de-fined as
 []

[104]
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[]\OT1/ptm/m/n/10.95 If \OT1/pcr/m/n/10.95 l=[(1,2),(1,2),()]\OT1/ptm/m/n/10.95
 , this is \OT1/pcr/m/n/10.95 AleshinGroup \OT1/ptm/m/n/10.95 ([][]10.1.14[][])
. More gen-er-ally, if
 []


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\OT1/pcr/m/n/10.95 l=[(1,2,(1,2),(),...,()] \OT1/ptm/m/n/10.95 has odd length, 
this is a free group of rank \OT1/pcr/m/n/10.95 Length(l)\OT1/ptm/m/n/10.95 , s
ee
 []


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[]\OT1/ptm/m/n/10.95 This is the first ex-am-ple of non-abelian free group. It 
is the group gen-er-
 []


Underfull \hbox (badness 10000) in paragraph at lines 6589--6591
\OT1/ptm/m/n/10.95 ated by \OT1/pcr/m/n/10.95 AleshinMachine([(1,2),(1,2),()])\
OT1/ptm/m/n/10.95 . It could have been de-fined as
 []


Underfull \hbox (badness 10000) in paragraph at lines 6589--6591
\OT1/pcr/m/n/10.95 FRGroup("a=\OML/cmm/m/it/10.95 <\OT1/pcr/m/n/10.95 b,c\OML/c
mm/m/it/10.95 >\OT1/pcr/m/n/10.95 (1,2)","b=\OML/cmm/m/it/10.95 <\OT1/pcr/m/n/1
0.95 c,b\OML/cmm/m/it/10.95 >\OT1/pcr/m/n/10.95 (1,2)","c=\OML/cmm/m/it/10.95 <
\OT1/pcr/m/n/10.95 a,a\OML/cmm/m/it/10.95 >\OT1/pcr/m/n/10.95 ")\OT1/ptm/m/n/10
.95 , but is rather de-fined us-ing
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[105]
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[106]
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[]\OT1/ptm/m/n/10.95 This is an in-fi-nite, 2-generated, tor-sion 3-group. It c
ould have been de-fined as
 []


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[107]
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[108]
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[]\OT1/ptm/m/n/10.95 It ad-mits an en-do-mor-phism on its de-rived sub-group. I
n-deed
 []


Underfull \hbox (badness 10000) in paragraph at lines 6899--6900
\OT1/pcr/m/n/10.95 FRElement(1,Comm(a,b))=Comm(c^-1,b/a)\OT1/ptm/m/n/10.95 , \O
T1/pcr/m/n/10.95 FRElement(1,Comm(a,c))=Comm(a/b,c)\OT1/ptm/m/n/10.95 ,
 []


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[109]
Underfull \hbox (badness 2894) in paragraph at lines 6911--6913
[]\OT1/ptm/m/n/10.95 This group was stud-ied in [[]][]. It is frac-tal, but not
 con-tract-ing. It could
 []


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\OT1/ptm/m/n/10.95 have been de-fined as \OT1/pcr/m/n/10.95 FRGroup("a=\OML/cmm
/m/it/10.95 <\OT1/pcr/m/n/10.95 ,b\OML/cmm/m/it/10.95 >\OT1/pcr/m/n/10.95 (1,2)
","b=\OML/cmm/m/it/10.95 <\OT1/pcr/m/n/10.95 a,c\OML/cmm/m/it/10.95 >\OT1/pcr/m
/n/10.95 ","c=\OML/cmm/m/it/10.95 <\OT1/pcr/m/n/10.95 a,a^-1\OML/cmm/m/it/10.95
 >\OT1/pcr/m/n/10.95 (1,2)")]]\OML/cmm/m/it/10.95 >\OT1/ptm/m/n/10.95 ,
 []


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\OT1/ptm/m/n/10.95 but is rather de-fined us-ing Mealy el-e-ments. It par-tiall
y ad-mits branch-ing on its
 []


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\OT1/ptm/m/n/10.95 sub-group \OT1/pcr/m/n/10.95 Subgroup(G,[a^2,(a^2)^b,(a^2)^c
])\OT1/ptm/m/n/10.95 , and, set-ting \OT1/pcr/m/n/10.95 x=Comm(a^2,b)\OT1/ptm/m
/n/10.95 , on
 []


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[110] [111]
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[112]
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[113]
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[114] [115] [116] [117] [118] [119] [120] [121]
Chapter 11.

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[122

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[123] [124] [125] [126] [127] [128]pdfTeX warning (ext4): destination with the 
same identifier (name{L.X7C73C74D87428A33}) has been already used, duplicate ig
nored
<to be read again> 
                   \relax 
l.8232 \hyperdef{L}{X7C73C74D87428A33}{}
                                         [129] [130]
Chapter 12.

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[131

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[]\OT1/ptm/m/n/10.95 The fourth and fifth com-mand com-press the newly cre-ated
 pe-ri-odic list, see
 []


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[132]
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(Font)              Font shape `OT1/pcr/b/n' tried instead on input line 8502.
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\OT1/ptm/m/n/10.95 If the first ar-gu-ment is an in-te-ger \OT1/pcr/m/n/10.95 n
 \OT1/ptm/m/n/10.95 and not a record, the com-mand is in-ter-preted as
 []


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[]\OT1/pcr/m/n/10.95 WordGrowth(...,rec(draw:=true)) \OT1/ptm/m/n/10.95 may be 
ab-bre-vi-ated as \OT1/pcr/m/n/10.95 Draw(...)\OT1/ptm/m/n/10.95 ;
 []


Underfull \hbox (badness 10000) in paragraph at lines 8526--8527
\OT1/pcr/m/n/10.95 WordGrowth(...,rec(ball:=n)) \OT1/ptm/m/n/10.95 may be ab-br
e-vi-ated as \OT1/pcr/m/n/10.95 Ball(...,n)\OT1/ptm/m/n/10.95 ;
 []

[134] [135] [136]
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[]\OT1/ptm/m/n/10.95 This func-tion con-structs an en-do-mor-phism of the group
,monoid or semi-
 []


Underfull \hbox (badness 10000) in paragraph at lines 8763--8764
\OT1/ptm/m/n/10.95 group [] spec-i-fied by send-ing gen-er-a-tor num-ber $\OML/
ptmcm/m/it/10.95 i$ \OT1/ptm/m/n/10.95 to the $\OML/ptmcm/m/it/10.95 i$\OT1/ptm
/m/n/10.95 th en-try in
 []


Underfull \hbox (badness 10000) in paragraph at lines 8763--8764
[]\OT1/ptm/m/n/10.95 . It is a short-cut for a call to \OT1/pcr/m/n/10.95 Group
HomomorphismByImagesNC \OT1/ptm/m/n/10.95 or
 []


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[]\OT1/ptm/m/n/10.95 This func-tion con-structs a ho-mo-mor-phism of the group,
monoid or semi-
 []


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\OT1/ptm/m/n/10.95 group [] spec-i-fied by send-ing gen-er-a-tor num-ber $\OML/
ptmcm/m/it/10.95 i$ \OT1/ptm/m/n/10.95 to the $\OML/ptmcm/m/it/10.95 i$\OT1/ptm
/m/n/10.95 th en-try in
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[]\OT1/ptm/m/n/10.95 . It is a short-cut for a call to \OT1/pcr/m/n/10.95 Group
HomomorphismByImagesNC \OT1/ptm/m/n/10.95 or
 []


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[137]
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[138]
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[139] (./fr.bbl (./fr.brf)
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