23. A. E. Zalesskii and V. N. Serezhkin, "Linear groups generated by transvections," Izv. Akad. Nauk SSSR, Ser. Mat., 40, No. i, 26-49 (1976).

PROBLEM OF FINITE AXIOMATIZABILITY FOR STRONGLY MINIMAL THEORIES OF

GRAPHS AND GROUPS WITH A NONZERO NUMBER OF ENDS

A. A. Ivanov UDC 510.67

This article is concerned with the problem of the existence of a finitely axiomatizable

strongly minimal theory of graphs of bounded valence, not categorical in countable power.

A positive solution of this problem would signify a major strengthening of a result of Pere-

tyat'kin [I], and a negative one would allow one to prove the absence of a quasivariety of

unary algebras finitely axiomatizable and not locally finite categorical in noncountable power

power [2].

The main result of this paper asserts that a strongly minimal theory (not categorical

in countable power) of graphs of bounded valence with a nonzero number of ends is not fi-

nitely axiomatizable. Hence, relying on [2], one can deduce, in particular, that finitely

generated infinite groups with a nonzero number of ends do not have a set of classes of con-

jugate elements such that each cyclic subgroup has a nontrivial intersection with one of these

classes.

Let us present the required definitions and preliminary information. By a graph we mean

a model of a finite number of binary predicates of signature d=<P/,,,°,~# . It is said

that a graph ~ of signature 6 has a valence no higher than S if for each ~, , /~L ~ ~, we s+/

have ~ (~I)(~ ~/2,'" {/~ +t) (/'~i (~,~'/)V~(~.,,2)) :o (~/ = ~e)). A complete

theory T of graphs has bounded valence if there exists a natural number $ such that all the

models of ~ have valence no higher than 8. All the graphs and their theories considered

subsequently have bounded valence. We call elements ~ and ~ from ~ directly connected (let

us denote =~ ) if =~ (=~ff~iV/(~i (Q,~ vPo (~,~))). we call ~ and ~ connected in

if there exist ¢ .... , Sf from ~ such that=~6, 4~CZ,.;.,O~ff. In this case, the se-

quence =,~,...,C~ff is called a path from ~ to if, and the number {+/ is called the length

of the given path. We call a maximal set /v of pairwise connected elements from ~ a com-

ponent of ~. If = and ~ are connected, then the distance ~(Q,b) between them is

If ~ and ~ are not connected, then we will let ~(~,~) -- ~ and for ~=~, we will let

d((~,~) = 0. For subsets I and C from S let us define the distance ~(~ C) between them:

d(~)=~f~[C/~,E) I ~E.~, Cg~). Let Q/,,,.,~/z be elements from s. Then we will call a

~-sphere of elements =,,,~,~Z~ , a submodel Sp(~al,...,~ ) of ~ with universe [~l(~/g)I~(al '

~) ~< ~ ) }. It is clear that in graphs of valence no higher than 8, only a finite number of

Translated from Algebra i Logika, Vol. 28, No. 2, pp. 160-173, March-April, 1989. Orig- inal article submitted September 30, 1987.

108 0002/5232/89/2802-0108512.50 © 1990 Plenum Publishing Corporation

types of an isomorphism of G-spheres of ~ elements, is realized. Let ~$./(t,~f,/,~g2,*.o,~ ), OS,2(~,a2.1,**',OZ, a ) , ,*, , 65, ~ ( t , ~ / , ~ . , a ~ , ~ ) be f ini te models of signature <~," ' ,P~,Q/ , ,,,, ~ • , being representatives of all possible types of the isomorphism of ~ -spheres

of constants O/, 0..,Q~ in graphs of valence no higher than $ . The formulation of the proposi-

tion "a ~-sphere of elements ~,.,., ~ has type ~S~ (~'~L,l'""~i,~) " is clear; we will call

the corresponding formula ~,~,~ [~,0..,~) a spherical formula.

Proposition 1 [3, 4]. a) If ~ is a complete theory of bounded valence, then each for-

mula of the language of ~ is equivalent to a Boolean combination of spherical formulas rela-

tive to ~.

b) For any natural numbers $ and ~ there are natural numbers ~f(8,~), Lz (8,N) such that

if the graphs ~ and ~ of valence no higher than $ for any type T of the isomorphism of

~I($~N) -spheres of one-element sets simultaneously contain an identical quantity of ~I{$,

~p-spheres of type ~', separated from each other by a distance no less than ~z (S,/D) or

simultaneously contain more than ~ such~/S,~) -spheres, then the graphs ~ and ~ satisfy

identical propositions containing no more than ~quantifiers (let us denote~ m ~ ).

Let us note here that assertion "b" is an immediate corollary of the proof of Lemma i

from [4].

Proposition 2 [3]. A complete theory ~ of bounded valence is ~-categorical if and only

if there exists a natural number ~ such that for any elements ~ and ~ of the model of 7", if

~(a, ~) ~ , then d(a,~) ~*

A component /~ of ~ of bounded valence is called uniform if for any elements ~,~E~

and any natural number ~ there exists an isomorphism ~=,~ from S~(N,~) to$~(~,~), taking

to~ . It is clear that in this case, the group of automorphisms of ~ acts on ~ transi-

tively (see [3, p. 53]). Furthermore, we have the next

Proposition 3. Let a strongly minimal theory ~ of bounded valence be not categorical

in countable power. Then

a) for any model ~ of ~ and for any ~ all the elements from~, with the exception of

a finite number, have identical ~-spheres;

b) there exists an infinite uniform component ~ such that certain models of ~ contain

infinitely many exemplars of~ ; moreover, such a 7" is unique for

From now on we will denote a component ~ from condition "b" as /~(~).

i. Graph Theories with a Nonzero Number of Ends

An infinite component /~ has a nonzero number of ends if there exists a natural number

and an element ~ from /" such that /"\$~) contains a minimum two infinite components.

THEOREM i. Let a uniform component ~ have a nonzero number of ends. Then for any /,i

there are a component /~' and an isomorphism ~ from ~ to such that /~ _=_ F , but

and F / are not elementarily equivalent graphs.

Proof. (For ease of comprehension, let us number the steps of the proof.) i) Let

be a natural number such that for some ~0 from ~ the graph~-$~(~) contains a minimum

two infinite components. Let us denote one of these components by~(Qo), and the other

Iz(~o). By ~{LT~) let us denote the subgraph /~,i<{S~)~SgIA~))[ If [ is some element

109

from F and c~ is an automorphism of /~ taking LT~ to { then we will let /7, ['~ = o~(,~(L7 o )) • , :. ,"

~Z (~)=~OC(/j(L20)), ~)(~ = oC(/~P(L~n }) . It is clear that one can represent 7 ~ in the form of

a disjunctive union of ~(~, ~(~), Pff).4C[ and;p U¢,{), where ~)and ~(~) are r7

infinite connected components of /",~(~d). Subsequently, for any ~ from / let us fix

the corresponding automorphism ~ and corresponding components ~[~), ~ ~]. Furthermore,

for any ~ and C from /v let us fix the automorphism oC{, C taking ~ to C, i/~{~ to ~ (C),

~:b to ~z(~} and, respectively, p(~) to P(O) [it is clear that C~, O takes 2/~(/~,~} tO sp (,,/, e ] .

Let us f i x t h e s e q u e n c e o f e l e m e n t s f rom //,C%' w h i c h a r e d i r e c t l y c o n n e c t e d

with elements from /v'/TldO). Let~/,,..,~/~ be a sequence of elements from peSO} directly connected with elements from /"" ~ :%} Let us assume that t 'q ~e ~ • I ~''*'~ are the corresponding

images from ~(~ of ~I,'*',~ with respect to the isomorphism ~ taking ~ to ~ , and ~P, ....

P be the corresponding images from ~{~) of ~! .... ~/O K •

2) Let us prove that for any natural nonzero number ~ , the graph ~ contains elements

and ~ such that the distance between Si(/~fl) and Sp(~,O ) is greater than 2~ + / , and

_ , , - , . f SpCN, either ~DI~]C~I (C) but the elements O. ~ do not enter in ~) for all &-- , or ~)--C~(O), but the elements ~; do not enter in P(~)for all ~_~/(.

First of all let us observe that for any ~I from /v there is in /~ an element ~ as

distant as one wants from ~z, such that ~(~,~t)C_~[~z) [respectively, one can find ~z such

that S~(~,~)C-~(4)]. This follows from the uniformity of P and from the fact that for

any ~Z£/" the component ~(~l) [respectively, ]~z {~2) ] contains an ~/-sphere of type ~p{~,~.), as far as one desires from Q. Let us also note that from now on, we will repeatedly use

the fact that the conditions 2~(~,I) C. /7(~) and ~(#,c)) ~ { entail {~f~,..; ~

~,..~II)-~q(C), and ~p{/~)C---~(C) and L~(~/2(A/,~),Sp(~/~))>~ entail {~f~,..,~ ~,.,,

Now let us assume that there do not exist elements ~ and C with the properties indi-

/ ' / 'S" ( , '~ I ~', ~ ' ' ~ I ~ / and cated above. Then if ~ and C. are elements from ~ such that ~ . ~.:, D[,4/,C})>. ~ fcP -~ Sp (~, I ) ~- ~(O) then p[~, C) ~ ! .... ,,.K' " 3 '-- -~ # . Since for any C~ -~" there exists a path

in the subgraph ,," \ ~p (A / ~] from

subgraph /"\ $~(~, ~.) from C to

# and SD(~,~)IT(,7(C)LISp(N,C)) =

P O~ to C and for any g from T, (C) there exists a path in the

P P [let us use the fact that 2D(/~ / ~)C~ iSD(NoC) U[C/....,~, ~ ) =

], then the sets ~[~) and ~7 (C) are connected by some

path inF\$p (~,~} . consequently, ~(C)c-/"(1), since the inverse contradicts the fact that

p[~) and ,/7, C~ are not connected in ~\ SO(J~ ~)'

Using the remark above, let us find an element e from /v such that0QiN, C ) c ~ -- ;l(~) ,

g~/iSo(~e) ,~/2(A<c)) >2~,+/ and di~(,4~e~,~i~(/~/,{))>2~ + / . Then according to the assump-

tion adopted;P.{C~,Q ,,..~F 7-~. As earlier, one can observe that in the subgraph / \

~ (~/,C)the sets ,/~, (C) and ~D (~/e)~ /! (C} and ~(e) are connected by certain paths. We

hence conclude that SpI,~)u~[~' c-~(c) Consequently, ~p~',E, L]~(e) CD(~) Since

ii0

(l,(..., ,:c.) C then[4: "",4"J OPC',: This co.- tradicts the assumption adopted.

3. In this manner, one can fix elements ~ and C from ~, such that one of the following

cases is satisfied:

a) ~(Sp{/~,fj, SpCN, c1)> ~+/, &f~,D=-:Tcc~, o/¢F,:::, i_~b-<:;

I f case "b" is s a t i s f i e d , then in the subgraph T,&:~D the component ' i (~5 contains

~P t~ &'+.K and, consequently, contains ~ (/4/,C.) and £ " ' ' ~ : i~: Since /~<Z~ f l ies in r'~(~)

then 8: "= 7 /7, lUJ, i -~ Z-~ : . Consequently, case "a" is s a t i s f i e d wi th the t ranspos i t ion

of ~ and C • In this manner, we will now consider that "a" is fulfilled for ~" and U - Then

we have the following: ~E~() , l~'~<f, ~p(<~) U:(C) c_/~7(() [the opposite signifies that in /"\$>(~,:) there exists a path from /7(~ to /71 (~ through some C, ~ ~ and ~(C) )

~(:)C#C~) [since ~E#(()O~C), /~&~, ~p(~,C) f~#(()= # and there does not exist in

/"%.2pl<c) a path from #(C) to ~(C) ].

4. Let ecc, ~ be the automorphism of ~ , defined at the start of the proof of the theorem,

and for each LZ from f through $(~2~ let us denote the subgraph ~ "~ (a) • For each integer

let us define a subgraph 4 and an isomorphism ~ ::~ --~). For Z---~ we will let 4--

~(:], and ~0 be the identity mapping. Let ~----- ~(O), and ~ be the restriction on ~(O) of

C~C,{. In view of ~/~,)C#(C) we have 4 6- 4" As 4 let us take o6"I C.~ (~)' and as ~Z , the

restriction of o~,~ on ~2" Let us remark that, moreover, ~z C 4 (since ~ C Co ), and one

can consider that 4=~(~2{ (()) [for this it is sufficient to select for o6~I~) a suitable

automorphism o~ from Sac. 1 of the proof]. Additionally, for each positiveS, we will let

~f= (q"l)' and ~" = cgC, e ~ be the restriction of ocC, op on ~'d" It is clear that =

O ( ¢lOa:/ { t~) ) and ~D,~q,_,. As C_/ let us take ~£,~(C(:) , and as ~9_1 let us take o~C.~ ~ ¢_,. Again it is simple to

' ~&).' understand that O 0 C uP./, and one can consider that 0. t = 0 [~C,# Now for each negative

we will let ~=o~C,~ (~+ !), and ~,=o~CZ~ <.. As a result we obtain two sequences: the sub-

graphs ,,.~4C~C4C~_! ~_2 C,,, and the isomorphisms ~&. , &'£ {0,+/,-+2,...} such that 0~=

:~ ' ¢- 1( , It is clear that the elements ~&. %/). Moreover, one can consider that &~'-~(~: :)).

directly with the elements from 4 (that is,~(OJL)L y,'"* ~J) are in ~0 • Therefore, for any

Z, the elements directly connected with elements from ~ , are in 4"4 " Using this and the

connectivity of /~, we obtain /~=.U= <-. From similar arguments we have ~--U~/f~I~'l[)). Conse-

quently, for any ~ from ~ there is a number D" such that ~ E ~, k q,,/,

Iii

5. Let us define a congruence ~ of /~ (in the sense of the definitions from [5]). We

,ITjV' < ~7/~O~((~Zl)), ~/C~)>E~"'))}, / ~ ~ . I t is i=ediate ly verif ied that ~(=~ is an equivalence relat ion on r - Let u~ show that for any ~ , we have ~ = e C ~ ) . Actua=y,

if ~ = / , g T z > E ~ / ' a n d O , ~ \ ~ + l , ~ZZE~* %~'+,, then either ~' ' / and in this case <~.~Z,), ~,.~;)> ~ r or ~+~=/ (the case /+~=~ is s imilarly considered). For ; + ~=¢" . e have <~.[~,),

[~Z )> ~ PT' since ~. is an isomorphism from ~ to CO. But in this case ~Z )~ ~/ and

~"+~ la~)= , ¢,. c~, ~, ~ , - ~ . c.~,) > ~ 5 f

I t remains to show that if <LZ6.,~[>E~(=), ~Et/ ,2] ,<" , ,~Z>~(~) , then <~,~2>68{~) .

Let us assume that gT/e-~'Of~et, (~zg~.'~/+/, ~/6-C~'4+ ' , ~zgCm"Cm+l, Then ~p~(g2l,=~'~} .

by this that g= <g(=),g{~),"',g~4 )> is a congruence of 77.

By 771 let us denote the factor-system /v/~, and by ~ the corresponding canonical homo-

morphism.

6. Now let us show that if distinct elements L71 and ~2 from /v are separated from each

other by a distance not exceeding 2Z+/, then ~(~I) # ~2 ((]z) • For this, let us observe that

if the distance from ~ to <. is no larger than ~ +/ , then ~& ~-_/ . Actually, if LI is not

in ~;_/ , then the shortest path from ~ to ~ must pass through $~ (~, ~i_/f ~ ()), and end at

$p(~fl[{)). In view of ~($~(~, ~;[l{~)) ) $p(/~71([]))~2.Z+/we obtain a contradiction with the fact that such a path has a length not exceeding 2~+/ .

In this manner, if ~{~21,ZTz)~ 2Z4/, then for some ~ either [al,LTzj _c~., ~f+l or LZ 1 &

~, k4.+l,~ 2 E~+/" ~'+Z" In the first case ~([/I) ~ ~}(~2 ), since ~O~l~./.} ~ ~ I~Z). In the

second case, the equality ~(Gf) = ~(Qz) entails ~. (=!) = ~'+I (~z), that is, the distance from

Sp(<~-l[{)) to ~, equals the distance from $~(~i//(~)) to L~ 2 . Furthermore, the shortest

path from ~, to ~2 must cross through ~(/~,~//{~), Therefore d(Q,,Q2) >~(O~$~(~, < ×

¢ ~ ) ) + d ( Sp ( a/, ¢7~ ' ( ~ ), a.z " ) = d ( a, , $p ( I¢, ~/+'! ~ [, > ) + d { Sp ( /¢, g i ' ¢ oe) ), cz , ) ._- d ( S ? ( N, ~a T/ { ~ ) ) ,

3p(,4/,90~1+1{~)))>2~+/, A contradiction.

7. Let us observe that for any ~ from C any two elements from $~{~, ~) are separated

from each other

beds ;p(Z~l~ ) in observe that if

ment ~3eF such

by a distance not exceeding 2L,. Therefore, the mapping ~ injectively em-

Sp(~,~(a)). To prove the fact that ~>I~,a) ~ SpI~,~[ Q)) it remains to

for some ~. and ~ fromF, we have /~f~(~(Ot),~(~21), then there is an ale-

that ~LZ",a~>~ ~(=) and /v~ '~'(~I,% ) Let us assume that g21~ 4- k4.+l ~ z .. ; "

, , then ~-f(~o(.z))6 ~ "~ 6+I and'G/,~J(~'IQzJ)>6 5 p

112

since ~' is an isomorphism from ~ to C 0, taking C~+ I into ~ . Therefore, as ~3 it is neces-

sary to take ~1(~/ (~2)). If <~[(~I),~;I(~.{~z))>6/ZT# P, then ~II~.(Q2~£ ~\~2 , and ~71(~ I ~.(~z~)~/\~2 In view of ~Z(~p-/ (~•(Q2) ~ = ~-! • l+/ ( ~/ (a£)). we have <~Z, ~d.! (~ (~)>&~(=), and since ~ is an isomorphism, then f~,~f~ (~.(~2))>6~ C. The case <~f[(~

(~, ~'(~Z)>E~T/ is considered similarly.

8. The end of the proof of the theorem. Let the valence of ~ not exceed $ . In the

2nd step of the proof, as ~ let us take(~+~(71($~)+Z 2 (S,~)) ; where the numbers L~C%~) and ~2(~ ) are defined in Proposition i. From the uniformity of ~ and Sac. 7 of the proof,

it is simple to obtain that the graphs ~ and / vl satisfy the conditions of Proposition 1

"b," that is, /"rag/"I. And what is more, /v and / vl are not elementarily equivalent, since I

for ff~>/+#~+ ~(~(~,~,&(~,C)) the graph f contains an fru-sphere not meeting /~. The

theorem is proved•

The theory [ is called finitely axiomatizable modulo positive propositions, if for some

finite set of formulas ~, the theory ~ is axiomatized by the set Z U 7 4 , where 2 -+ is the

set of all positive corollaries of ~ .

THEOREM 2. Let a strong minimal theory ~ of bounded valence not be categorical in

countable power• Then if/v(~) has a nonzero number of ends, then [ is not finitely axi-

omatizable modulo positive propositions.

Proof. Let the graph ~ ~ contain /"(~) . By Theorem i for any ~ one can find a homo-

, '-- /7(r) but /,i and /"(rJ are not elementarily morphic image ~' of /~(~) such that ~----~

equivalent. Replacing in LT~ the component /"(~) by ~', we obtain a homomorphic image ~i of

~ such that~ ~/ From the strong minimality of ~(~) follows the noncoincidence of

~(~] and ~[~ (~ and ~' contain a distinct number of F/Z-spheres of a certain type ~',

meeting a finite number of times in ~). To conclude the proof of the theorem, it remains

to recall that the positive propositions are stable with respect to the homomorphisms.

Remark Io An immediate corollary of Theorem 2 is Theorem 7 from [3], according to which

the theory of a uniform not 2-connected component is not finitely axiomatizable. And what

is more, if the lengths of the cycles in the uniform component ~ are bounded, then Theorem 2

entails the not finite axiomatizability of the theorem of the graph 2".

Remark 2. In the conditions of the proof of Theorem 2, let the basic predicates ~ ,

/&~ ~ ~, yield in the algebraic system ~ the local functions ~ (X)= ~ q--~;.~7 Z (~,~). Then

according to Sac. 7 of the proof of Theorem i, the component /Viand the algebraic system ~'

are unary algebras with respect to the functions similarly defined in them, and the homo-

morphism from ~ to $~! is a homomorphism of unary algebras• Hence, in particular, it follows

that for a theory [ of unary algebras, the proof of Theorem 2 goes through too in the case

of a functional signature.

2. Groups with a Nonzero Number of Ends

Let ~(K) be the class of unary algebras of signature ~=<~,,.,,~, ~,,..,,~K > ' satis-

fying the following axioms:

113

2) ~(~)~)(~C~)--~ ~-~C~)=~), ~ is a term of signature ~.

<. ~ /-~ i~ K, as binary predicates It is clear that if one considers the operations ' i '

then the unary algebras from ~(/~) are graphs of bounded valence. Moreover, all the termin-

ology and results mentioned in the introduction are carried over to the case of algebras from

LEMMA i. For any unary algebra ~ from $CK) , the elementary theory ~/~) is strongly

minimal and allows the elimination of quantifiers.

Proof. In view of Proposition i, "a," it is sufficient to observe for the proof of

eliminability of the quantifiers that each spherical formula is equivalent to a quantifier-

less formula. By the axiom 1 of class ~(K), the spherical formula ~/,i,~ (~f~ "°'~R ) is

equivalent to the conjunction of equalities and inequalities of the form ~=~2 , ~ ~ ~ '

where ~ , ~z ' ~ ' ~ are terms of signature ff of the variables from ~,,..1~"tz} containing

not more than ~ functional symbols.

Now, if ~,(ZI~...,~2 ~ ) has infinitely many realizations, then, without destroying

generality, one can consider that ~,~ZI~...~) is the conjunction of atomic formulas and

their negations in the signature ~. Let us remark that ~(X,~ ..... ~ ) does not contain

equations of the form ~t (~}-~(Zi),/~6~/Z, since otherwise by axiom i, the formula ~,

a#~'." ~)has no more than one realization. Therefore, if as the number ~ one takes

~a~ ~/?1 +~Z ~fZI~/Z; are corresponding lengths of the terms ~ (~ , ~2~ ~ from ~(~) ~z~ ~ ~ )

in ~(.f'~2#,o,.,~2 n }) + {, then by axiom 2 each element separated from [~..,,~a~ by a distance

not less than/~ , satisfies ~(X,~Yl,..0,~n ). The lemma is proved.

Let ~,~l,...,~Z~> be an arbitrary group with distinguished elements ~...,/7 K . Let us

construct a unary algebra ~/~(<~,~f,.,, ,~21<>) of signature ~ in the following manner: as the

universe of the algebra let us take the set ~ and define on it the operations ~,(m)= gT? ~

~ (~2)=~i{,f , ~6..~K. It is easy to understand that the algebra ~/g(~,~l~ ...,~) is in

~[K]. If the group ~ has a representation <L7/,,o.,6[~ ; E ~, .... ~)>, where Z ~LT,,.,,, LT~ )

is the set of words with generators ~/~,.., LT~, then the unary algebra ~IZ(~/,.,.~#) actu-

ally represents the graph of ~ and is the free 1-generated algebraic variety V~I(,~),

given by the identities:

Z .... ,a,<

where the term WID,.°.,/~,~/,...,4~)(~} is obtained from W{4,.°.,(Z ~} by replacing each occurrence

of ~. by /z" and each occurrence of ~/ by ~., /-~'~ ~ . This is immediately verified.

Let us remark that all the connected unary algebras from ~(K) can be obtained in the

manner described above. Actually, if the connected algebra ~ is in ~{K) , then let us choose

an arbitrary element gZ from ~. Then each element ~ from ~ can be represented in the form

#(ZZ) , where ~) is some term of signature ~. Using axioms 1 and 2, it is simple to

114

! understand that ~ is a group with respect to the natural multiplication ~IL2)' ~ (a) =

(~I~£2)). We will denote the group obtained by G~; it is generated by k~),,,o,/K ~Q).

We obtain by the direct verification ~/ZI~Oy,//~L2),,,,~D (LZ)>)=~,

Remark. It is necessary to note the obvious connection of this material with the graphs

of the groups [6, p. 70].

LEMMA 2. If the components of the unary algebra ~ from ~) are infinite, then there

exists a finitely generated group ~ with generators 0/ .... ~ ~Z~ such that ~,(<~!,.0.,

Proof. It is simple to understand that for any unary algebra ~ from ~{~) and for any

~I and ~z from ~ there exists an isomorphism from the component 17 containing 4 ' to the com-

ponent /~z containing {z" This isomorphism takes ~ to ~z. For this, in view of axioms i and

2, it is sufficient for each term ~(~) of signature ~ to take ~[4] to #(4). In this

manner, the unary algebra ~ consists of isomorphic copies of some component ~. Using

Proposition 1 and the infinitude of ~, we obtain ~m/" . Now as ~ it is sufficient to

take ~/ . The lemma is proved.

The next theorem is a version of a theorem of Taitslin and Shishmarev from [2].

THEOREM 3. Let the infinite group $ with the representation <L~I~,,. , O K ; Z(L7/ .... ,~K)>

[here ~ (L71,,,,~LZ ~ ) can be infinite] contain a finite set of classes of conjugate elements

such that each cyclic subgroup has a nontrivial intersection with one of these classes. Then

the elementary theory ~I~/Z (~.g~/, ,,.,aK>)) is finitely axiomatizable modulo positive propo-

sitions.

Let us make a small remark in connection with the proof of Theorem 3. Let V4(¢, .... ~b~),

..., Vs~L1,;..,~b~) be the representatives of those classes from ~, about which it was

spoken in the formulation. Then, the axioms [~/~/(<~4~...j O~m>)) are the following propo-

sitions :

VZ Cp....,/~, ~,, .... 7~<)(~) # m, t,~Z~ 5,

where the terms ~/l~.,..o.,,f,,~,q,,...~,q.)(',2~.~ [respectively, Vii ~ .... 'D' ~'~"''~'. )(~)] are obtained

from the words W(LT/,.,,,~/~)[respectively, ~. (Q~ .... , ~ )] in the natural manner. The proof of

this fact is a literal repetition of the proof from [2].

Let us move on to the main result of the section. It is said that a finitely generated

group ~ has a nonzero number of ends (see [7, 8, p. 312]) if the graph of ~ has a nonzero

number of ends. In particular, in this case ~ I<$,~2!,,,,, ~>), where ~'~, .... g7 K are the

generators of ~ , has a nonzero number of ends.

THEOREM 4. A finitely generated infinite group ~ with a nonzero number of ends does

not contain a finite set of classes of conjugate elements such that each nontrivial cyclic

subgroup has a nontrivial intersection with one of these classes.

115

Theorem 4 is an immediate corollary of Theorem 3, Lemma i, Theorem 2 (see Remark 2)

and the fact that

r'(rk.(/.z,z(<Q, (see the proof of Lemma 2).

From the description of J. Stallings of groups with an infinite number of ends, we

obtain

COROLLARY. If a finitely generated infinite group ~ is representable in the form of an

~ -extension with finite connected subgroups, or in the form of a nontrivial free prod-

uct with finite united subgroups, then G satisfies the conclusion of Theorem 4.

Let us remark that the finiteness condition of the connected or united subgroups in the

given corollary can be omitted [9]. In particular, finitely generated infinite groups with

one determining relation satisfy the conclusion of Theorem 4. Furthermore, Theorem 4 can

be proved by purely group-theoretic methods, using the Stallings theorem.

The author considers the fact interesting that in the case given, the use of a classi-

cal result from the theory of groups (Stallings' theorem) can be eliminated using a result

from mathematical logic (Theorem 2).

Comment in Proof. In connection with Theorems 2 and 4 of this paper, let us note the

result of S. V. Ivanov on the existence of a fin.itely generated infinite group with a finite

number of classes of conjugate elements (S. V. Ivanov, "Three theorems on the embedding of

groups," llth All-Union Symp. on Group Theory, Sverdlovsk, 1989, 51). From it, in particular,

the existence follows of a theory of graphs of bounded valence, strongly minimal and finitely

axiomatizabile modulo positive propositions (we apply our Theorem 3).

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