On relative hyperbolicity for a group and relative quasiconvexity for a subgroup

0.79 MB·English

by  Yoshifumi Matsuda

ON RELATIVE HYPERBOLICITY FOR A GROUP AND
RELATIVE QUASICONVEXITY FOR A SUBGROUP
YOSHIFUMIMATSUDA,SHIN-ICHIOGUNI,ANDSAEKOYAMAGATA
3
1
Abstract. Weconsidertwofamiliesofsubgroupsofagroup. Eachsubgroup
0
which belongs to one family is contained in some subgroup which belongs to
2
the other family. We then discuss relations of relative hyperbolicity for the

n groupwithrespecttothetwofamilies, respectively. Ifthegroupissupposed
a to be hyperbolic relative to the two families, respectively, then we consider
J relations of relative quasiconvexity for a subgroup of the group with respect

7 tothetwofamilies,respectively.
1

]
R
1. Introduction
G
. Relative hyperbolicity for groups is a generalization of hyperbolicity for groups
h
t (see [6]). There exist several definitions of relative hyperbolicity for groups which
a
are mutually equivalent for finitely generated groups ([1], [4], [5], [7] and [11]).
m
Relative quasiconvexity for subgroups is defined and studied by many authors (see
[ for example [7]).

1 The aim of this paper is to extend some known results on relative hyperbolicity
v for groups and on relative quasiconvexity for subgroups to one of the most general
9 cases. Indeed, we adopt Osin’s definition [11, Definition 2.35] (see Subsection 2.1)
2
of relative hyperbolicity for a group. For a definition of relative quasiconvexity for
0
a subgroup, [10, Definition 1.2] (see Definition 2.3 and Remark 2.5) is adopted.
4
. Throughout this paper, we assume that groups are not necessarily countable.
1
Unless otherwise stated, Λ denotes a set which is not necessarily countable and for
0
each λ Λ, we denote by M a set which is also not necessarily countable.
3 λ
∈
1 Let G be a group and let Hλ = Hλ λ Λ be a family of subgroups of G.
{ } { | ∈ }
: For each λ Λ, let K = K µ M be a family of subgroups of H .
v ∈ { λ,µ}µ { λ,µ | ∈ λ} λ
i We set Kλ,µ = Kλ,µ λ Λ,µ Mλ .
X { } { | ∈ ∈ }
One of our main theorems is the following on relative hyperbolicity for a group.
r
a Theorem 1.1. Let G be a group and H a family of subgroups of G. For each
λ
{ }
λ Λ, let K be a family of subgroups of H . Then the following conditions
λ,µ µ λ
∈ { }
are equivalent:
(i) G is hyperbolic relative to H and K , respectively.
λ λ,µ
{ } { }
(ii) G is hyperbolic relative to K and the following conditions hold:
λ,µ
{ }
2010 Mathematics Subject Classification. 20F67,20F65.
Key words and phrases. Relativelyhyperbolicgroups,relativelyquasiconvexsubgroups.
ThefirstauthorissupportedbytheGlobalCOEProgramatGraduateSchoolofMathematical
Sciences,theUniversityofTokyo,andGrant-in-AidforScientificResearchesforYoungScientists
(B)(No. 22740034),JapanSocietyofPromotionofScience.
ThesecondauthorissupportedbyGrant-in-AidforScientificResearchesforYoungScientists
(B)(No. 24740045),JapanSocietyofPromotionofScience.
1
2 Y.MATSUDA,S.OGUNI,ANDS.YAMAGATA
(1) G satisfies Condition (a) with respect to H (see Definition 2.12).
λ
{ }
(2) H is almost malnormal in G (see Subsection 2.1).
λ
{ }
(3) For any λ Λ, the group H is quasiconvex relative to K in G.
λ λ,µ
∈ { }
(iii) G is hyperbolic relative to H and for any λ Λ, the group H is
λ λ
{ } ∈
hyperbolicrelativeto K . ThereexistsafinitesubsetΛ ofΛsuchthat
λ,µ µ 1
{ }
for any λ Λ Λ , the subgroup H is the free product H = K .
∈ \ 1 λ λ ∗µ∈Mλ λ,µ
WhenΛisfinite, Condition(ii)(resp.(iii))isreplacedwiththefollowingcondition
(ii)’ (resp. (iii)’):
(ii)’ G is hyperbolic relative to K and the following conditions hold:
λ,µ
{ }
(2) H is almost malnormal in G.
λ
{ }
(3) For any λ Λ, the group H is quasiconvex relative to K in G.
λ λ,µ
∈ { }
(iii)’ G is hyperbolic relative to H and for any λ Λ, the group H is
λ λ
{ } ∈
hyperbolic relative to K .
λ,µ µ
{ }
Remark 1.2. If G is countable, and both Λ and M are finite, then the
λ Λ λ
equivalence between Conditions (i) and (ii)’ in Theore∈m 1.1 is proved by Yang
(cid:70)
[14, Theorem 1.1] and the implication from (i) (and (ii)’) to (iii)’ follows from
[14, Theorem 1.1 and Corollary 3.5]. If G is finitely generated, and both Λ and
M are finite, Dru¸tu and Sapir show that Condition (iii)’ in Theorem 1.1
λ Λ λ
imp∈lies Condition (i) in Theorem 1.1 (see [4, Corollary 1.14]).
(cid:70)
IfweomittheconditionthatΛisfinite,thentheimplicationfrom(ii)’to(i)(resp.
from(ii)’to(iii)’)inTheorem1.1isnottrue. Thefollowingisanexample. Foreach
l N 0 , let C and C be groups isomorphic to Z and let C be a group
3l+1 3l+2 3l
is∈omo∪rp{hic}to Z/2Z. For each m,n N 0 , set K =C C , K =
2m 3m 3m+1 2m+1
∈ ∪{ } ∗
C3m+2 and Hn = C3n C3n+1 C3n+2 C3(n+1). We put G = n N 0 Kn. The
∗ ∗ ∗ ∗ ∈ ∪{ }
group G is thereby hyperbolic relative to Kn n N 0 . The family Hn n N 0
is almost malnormal in G and for any n {N } ∈0∪,{th}e group H is{qua}sic∈on∪v{ex}
n
∈ ∪{ }
relative to Kn n N 0 in G. However G does not satisfy Condition (a) with
respect to {{Hn}}n∈∈N∪∪{{0}}because of Hn∩Hn+1 = C3(n+1) ∼= Z/2Z. It follows that
G is not hyperbolic relative to Hn n N 0 .
{ } ∈ ∪{ }
We give an example such that if one omits the condition that Λ is finite, then
theimplicationfrom(iii)’to(i)(resp.from(iii)’to(ii)’)inTheorem1.1isnottrue.
Foreachm N 0 ,letC beagroupisomorphictoZ/2ZandsetK =Z C
m 2m m
aisndcleKar2mth+a1t∈=GZ∪is×{hC}ympe.rbWoelicpuretlaHtmive=toK2mHm∗CmmKN2m0+1aannddfGor=an∗ymm∈N∪{0N}H×m.0It,
{ } ∈ ∪{ } ∈ ∪{ }
the group H is hyperbolic relative to K ,K . However G does not satisfy
m 2m 2m+1
{ }
Condition (a) with respect to Kn n N 0 . The group G is hence not hyperbolic
{ } ∈ ∪{ }
relative to Kn n N 0 .
{ } ∈ ∪{ }
The following is another our main theorem on relative quasiconvexity for a sub-
group.
Theorem 1.3. Suppose that G is hyperbolic relative to H and also hyperbolic
λ
{ }
relative to K . Then for any subgroup L of G, the following conditions are
λ,µ
{ }
equivalent:
(i) L is quasiconvex relative to K in G.
λ,µ
{ }
(ii) Lisquasiconvexrelativeto H inG. Foranyg G,thefamily gLg 1
λ −
{ } ∈ { ∩
H of subgroups of G is uniformly quasiconvex relative to K in
λ λ Λ λ,µ
} ∈ { }
G (see Definition 2.4 and Remark 2.5).
ON RELATIVE HYPERBOLICITY FOR A GROUP 3
(iii) L is quasiconvex relative to H in G. For any λ Λ and g G, the
λ
{ } ∈ ∈
group gLg 1 H is quasiconvex relative to K in H . For each
− λ λ,µ µ λ
∩ { }
g G, there exists a finite subset Λ of Λ satisfying the following: For
g
∈
any λ Λ Λ , the group gLg 1 H is decomposed into a free product
g − λ
∈ \ ∩
gLg 1 H = (gLg 1 K ).
− ∩ λ ∗µ∈Mλ − ∩ λ,µ
WhenΛisfinite, Condition(ii)(resp.(iii))isreplacedwiththefollowingcondition
(ii)’ (resp. (iii)’):
(ii)’ L is quasiconvex relative to H in G. For any λ Λ and g G, the
λ
{ } ∈ ∈
group gLg 1 H is quasiconvex relative to K in G.
− λ λ,µ
∩ { }
(iii)’ L is quasiconvex relative to H in G. For any λ Λ and g G, the
λ
{ } ∈ ∈
group gLg 1 H is quasiconvex relative to K in H .
− λ λ,µ µ λ
∩ { }
Remark 1.4. If G is countable, and both Λ and M are finite, Yang proves
λ Λ λ
that Condition (i) is equivalent to Condition (ii)’ (se∈e [14, Theorem 1.3]) and it is
(cid:70)
proved that Condition (ii)’ is equivalent to Condition (iii)’ in [9, Proposition 5.1].
We note that (ii)’ and (iii)’ in Theorem 1.3 are equivalent even if Λ is not finite
(see Proposition 3.3). We give an example such that if we omit the condition that
Λ is finite, then the implication from (iii)’ to (i) in Theorem 1.3 is not true.
For each n N 0 , let K be an infinite cyclic group generated by a . For
n n
each m N ∈0 ,∪w{e p}ut Hm = K2m K2m+1 and G = n N 0 Kn. It is clear
∈ ∪{ } ∗ ∗ ∈ ∪{ }
that G is hyperbolic relative to Hm m N 0 (resp. Kn n N 0 ) and for each
m N 0 , the group H is h{yper}bo∈lic∪{re}lative to{ K} ∈,K∪{ } . For each
m 2m 2m+1
m ∈ N∪{0}, we consider an infinite cyclic subgroup{L of H g}enerated by
m m
a ∈a ∪.{F}or each m N 0 , the subgroup L is then quasiconvex relative
2m 2m+1 m
∈ ∪{ }
to K2m,K2m+1 in Hm. We put L = m N 0 Lm which is a subgroup of G.
{ } ∗ ∈ ∪{ }
The group L is quasiconvex relative to Hm m N 0 in G. However L is not
{ } ∈ ∪{ }
quasiconvex relative to Kn n N 0 in G. In fact, if L was quasiconvex relative
{ } ∈ ∪{ }
to Kn n N 0 in G, then L would be hyperbolic relative to , that is, L would
be{a hy}pe∈rb∪o{lic}group by Theorem 4.25 in [10] because for ever∅y n N 0 and
∈ ∪{ }
every g G, the group L gK g 1 is trivial. This contradicts the fact that all
n −
∈ ∩
hyperbolic groups are finitely generated.
This paper is organized as follows. In Section 2, basic terminology is provided
andweshowLemma2.8onminimalliftsofquasigeodesicswhichisakeylemmain
thispaper. InSection3, severalpropertiesofrelativequasiconvexityforsubgroups
are introduced and Theorem 1.3 is proved. In Section 4, we prove Theorem 1.1.
2. Preliminaries
In Subsection 2.1, we recall Osin’s definition of relative hyperbolicity for groups
and a definition of relative quasiconvexity for subgroups ([10, Definition 1.2]). In
Subsection 2.2, a minimal lift of a path is defined and some properties are studied
by using minimal lifts. In Subsection 2.3, we recall the definition of Condition (a).
We fix some notations in this paper.
Notation 2.1. Let G be a group which is not necessarily countable. Let H be
λ
{ }
afamilyofsubgroupsofG. Foreachλ Λ, K denotesafamilyofsubgroups
λ,µ µ
∈ { }
of H .
λ
4 Y.MATSUDA,S.OGUNI,ANDS.YAMAGATA
For each λ Λ, put =H 1 and = . For each λ Λ, we set
∈ Hλ λ\{ } H λ ΛHλ ∈
= (K 1 ) and = . We re∈gard , , and as sets
Kofλletterµs.∈Mλ λ,µ\{ } K λ∈ΛKλ (cid:70) Hλ H Kλ K
(cid:70) (cid:70)
2.1. Notation and terminology. We refer to [11] for details. We say that G is
generated by a set X relative to H if G is generated by X . In this case, the
λ
{ } (cid:116)H
setX issaidtobearelativegeneratingsetofGwithrespectto H . Inthispaper,
λ
{ }
we assume that a relative generating set is symmetric. If X is finite, we say that G
is finitely generated by X relative to H and X is a finite relative generating set
λ
{ }
of G with respect to H . If there exists a finite relative generating set of G with
λ
{ }
respect to H , we simply say that G is finitely generated relative to H .
λ λ
{ } { }
TheinclusionfromH intoGinducesthehomomorphismfromthefreeproduct
λ
F =F(X) ( H ) onto G, where F(X) is the free group with the basis X. Let
λ Λ λ
∗ ∗ ∈
N be the kernel of this homomorphism. We take a set consisting of words over
R
X . If N is equal to the normal closure of a subset in F whose elements are
(cid:116)H
exactly represented by elements in , then
R
(2.1) X,H ,λ Λ R=1, R
λ
(cid:104) ∈ | ∈R (cid:105)
is called a relative presentation of G with respect to H . In this paper, is
λ
{ } R
assumedtobesymmetricandtocontainallcyclicshiftsofanyelementsof . IfG
R
hasarelativepresentation(2.1)withrespectto H ,andbothX and arefinite,
λ
{ } R
then the relative presentation (2.1) with respect to H is called a finite relative
λ
{ }
presentation of G with respect to H . If there exists a finite relative presentation
λ
{ }
of G with respect to H , the group G is said to be finitely presented relative to
λ
{ }
H .
λ
{ }
The relative presentation (2.1) of G with respect to H is said to be reduced
λ
{ }
if each R of has minimal word length among all words over X representing
R (cid:116)H
the same element of the group F. Unless otherwise stated, we assume that every
relative presentation of a group with respect to a family of subgroups of the group
is reduced.
We assume that G has the relative presentation (2.1) with respect to H . Let
λ
{ }
W be a word over X . The word length of W is denoted by W . Suppose
(cid:116)H (cid:107) (cid:107)
that W represents the neutral element 1 in G. The word W is then denoted by
W = ni=1fi−1Rifi in F, where each i = 1,2,...,n, fi ∈ F and Ri ∈ R. We
denote by Arearel(W) the smallest number n among all such representations of
(cid:81)
W as above and call it the relative area of W with respect to (2.1). A function
f: N N is called a relative isoperimetric function of G with respect to (2.1) if f
satisfi→es the following: For any n N and any word W over X representing
∈ (cid:116)H
theneutral element1 in G with W n, theinequality Arearel(W) f(n)holds.
(cid:107) (cid:107)≤ ≤
The smallest relative isoperimetric function of G with respect to (2.1) is said to
be the relative Dehn function of G with respect to (2.1). We say that the relative
Dehn function δ of G with respect to (2.1) is well-defined if for each n N, the
∈
value δ(n) is finite. The relative Dehn function is not necessarily well-defined (see
[11, Example 2.55] or [12, p.100]).
WesaythatGishyperbolicrelativeto H ifGhasafiniterelativepresentation
λ
{ }
with respect to H and the relative Dehn function of G with respect to the
λ
{ }
presentationislinear. Thisdefinitionisindependentofthechoiceofafiniterelative
presentation of G with respect to H (see [11, Theorem 2.34]).
λ
{ }
ON RELATIVE HYPERBOLICITY FOR A GROUP 5
LetΓbeagraphwiththesetV (resp.E)ofvertices(resp.edges). Inthispaper,
a graph means an oriented graph, i.e., every edge is oriented. For each edge e E,
∈
we denote the origin and the terminus of e by e and e , respectively. The inverse
+
edge of e is denoted by e 1. Note that (e 1) =− e and (e 1) =e .
− − + − +
− −
Foreachi=1,2,...,n,lete beanedgeofΓ. Letp=e e e beasequenceof
i 1 2 n
···
edges satisfying that for any i=1,2,...,n 1, the equality (e ) =(e ) holds.
i + i+1
The sequence p is called a path from (e ) −to (e ) in Γ. The vertices (e−) and
1 n + 1
− −
(e ) are denoted by p and p , respectively. Assuming that the length of every
n + +
−
edge of Γ is equal to 1, we regard Γ as a metric space. The length of p is denoted
by l(p). For a path p = e e e , we express a path (e ) 1 (e ) 1(e ) 1 by
1 2 n n − 2 − 1 −
··· ···
p 1. When p =p , the path p is called a cycle in Γ.
− +
Let u and−v be vertices of Γ. A geodesic from u to v in Γ is a path from u to v
in Γ whose length is minimal in the set of all paths from u to v in Γ.
Let X be a relative generating set of G with respect to H . We define the
λ
{ }
CayleygraphΓ(G,X )ofGasagraphsuchthatthesetofvertices(resp.edges)
(cid:116)H
is equal to G (resp. G (X )) and each edge (g,u) G (X ) satisfies that
× (cid:116)H ∈ × (cid:116)H
(g,u) =g and (g,u) =gu.
+
Th−elabelofeachedgee=(g,u) G (X )isdenotedbyu=φ(e). Thelabel
∈ × (cid:116)H
oftheinverseedgee 1 ofeisu 1 =φ(e 1)=φ(e) 1. Forapathp=e e e in
− − − − 1 2 n
···
Γ(G,X ), the label of p is defined as φ(e )φ(e ) φ(e ) and denoted by φ(p).
1 2 n
(cid:116)H ···
For each path p, we denote by φ(p) the element of G represented by φ(p).
We recall the definition of Condition (b) introduced in [10, Definition 1.1].
Definition 2.2. Let L be a subgroup of G. The subgroup L is said to satisfy
Condition (b) with respect to H if for any y ,y G with Ly Ly = , the
λ 1 2 1 2
subset H(L,y ,y )= H λ{ Λ},y H Ly = ∈of H is fini∩te. ∅
1 2 λ 1 λ 2 λ
{ | ∈ ∩ (cid:54) ∅} { }
Let Y be a subset of G. For any g,h G, we put
∈
dY(g,h)=min{ n|g =hy1y2···yn,i=1,2,...,n,yi ∈Y or yi−1 ∈Y}.
If there exist no elements yi ∈ Y nor yi−1 ∈ Y with g = hy1y2···yn, then we set
d (g,h)= . For any g,h G, we also denote d (g,h) by h 1g .
Y Y − Y
∞ ∈ | |
For subsets Y, A and B of G, put
d (A,B)=min d (a,b) a A,b B .
Y Y
{ | ∈ ∈ }
Let G be a group which is hyperbolic relative to H . When G is finitely
λ
{ }
generatedandΛisfinite,OsindefinesquasiconvexityforsubgroupsofGrelativeto
H ([11, Definition 4.9]). We recall [10, Definition 1.2] which is a generalization
λ
{ }
of the definition when both G and Λ are not necessarily countable. If G is finitely
generated and Λ is finite, [10, Definition 1.2] is equal to [11, Definition 4.9].
Definition 2.3. Let G be a group which is hyperbolic relative to H and let X
λ
{ }
be a finite relative generating set of G with respect to H . A subgroup L of G is
λ
{ }
said to be pre-quasiconvex relative to H in G with respect to X if there exists a
λ
{ }
finite subset Y of G satisfying the following: Let l be an element of L and let p be
a geodesic from 1 to l in Γ(G,X ). For each vertex v of p, there exists a vertex
(cid:116)H
w of L such that d (v,w) 1.
Y
≤
If L is pre-quasiconvex relative to H in G with respect to X and satisfies
λ
{ }
Condition (b) with respect to H , the group L is said to be quasiconvex relative
λ
{ }
to H in G with respect to X.
λ
{ }
6 Y.MATSUDA,S.OGUNI,ANDS.YAMAGATA
ForafamilyofsubgroupsofagroupG,wedefineuniformrelativequasiconvexity.
Definition 2.4. Suppose that G is hyperbolic relative to H and that X is a
λ
{ }
finite relative generating set of G with respect to H . Let L be a family
λ ν ν N
of subgroups of G. We say that L is uniformly{qua}siconve{x re}lat∈ive to H in
ν λ
{ } { }
G with respect to X if it holds the following conditions:
For any ν N, the group L satisfies Condition (b) with respect to H .
ν λ
• ∈ { }
ThereexistsafinitesubsetZ ofGsatisfyingthefollowing: Takeanyν N
• ∈
andanygeodesicpinΓ(G,X )whoseoriginandterminuslieinL . For
ν
(cid:116)H
any vertex u of p, there is a vertex v of L with d (u,v) 1.
ν Z
≤
Remark 2.5. One can show that the above definitions are independent of the
choiceofafiniterelativegeneratingsetofGwithrespectto H byusingrelative
λ
{ }
hyperbolicity (see [10, Corollary 4.21]). The subgroup L is thus simply said to be
pre-quasiconvex (resp. quasiconvex) relative to H in G if there exists a finite
λ
{ }
relativegeneratingsetX ofGwithrespectto H suchthatLispre-quasiconvex
λ
{ }
(resp. quasiconvex) relative to H in G with respect to X. The family L
λ ν ν N
isalsosimplysaidtobeuniform{ly q}uasiconvex relative to H in Gifthe{ree}xi∈sts
λ
{ }
a finite relative generating set X of G with respect to H such that L is
λ ν ν N
{ } { } ∈
uniformly quasiconvex relative to H in G with respect to X.
λ
{ }
We recall the definition of almost malnormality. The family H is said to be
λ
{ }
almost malnormal in G if
for any distinct λ ,λ Λ and any g G, the group gH g 1 H is
• 1 2 ∈ ∈ λ1 − ∩ λ2
finite; and
for any λ Λ and any g G H , the group gH g 1 H is finite.
λ λ − λ
• ∈ ∈ \ ∩
If G is hyperbolic relative to H , then H is almost malnormal in G by [11,
λ λ
{ } { }
Proposition 2.36].
2.2. Minimal lifts of quasigeodesics. Let G be a group with the relative pre-
sentation (2.1) with respect to H . For a word W over X and each λ Λ,
λ
{ } (cid:116)H ∈
a non-trivial subword V of W is called an H -subword of W if V consists of letters
λ
from . We say that an H -subword of W is an H -syllable if it is not contained
λ λ λ
H
in any longer H -subword of W. Let p be a path in Γ(G,X ). For each λ Λ,
λ
(cid:116)H ∈
we say that a subpath q of p is an H -component of p if φ(q) is an H -syllable of
λ λ
φ(p). A subpath q of p is also said to be an -component of p if there exists an
H
element λ of Λ such that q is an H -component of p.
λ
Let p be a path in Γ(G,X ) and for any λ Λ, let q and r be non-trivial
(cid:116)H ∈
subpaths of p labeled by words over . If there exists a path c in Γ(G,X )
λ
H (cid:116)H
from a vertex of q to a vertex of r which is labeled by letters from , then q and
λ
H
r are said to be connected and c is called a connector. We permit the case where c
is trivial. If an H -component q of p is not connected to any other H -component
λ λ
of p, then we say that q is isolated.
Foreachλ Λ,letgbeanelementof suchthatthereexistsanH -syllableof
λ λ
∈ H
someR whichrepresentsg inG. WedenotebyΩ thesetofallsuchelements
λ
∈R
as g and put
λ
∈H
(2.2) Ω= Ω .
λ
λ Λ
(cid:71)∈
ON RELATIVE HYPERBOLICITY FOR A GROUP 7
Note that for each λ Λ, the set Ω is symmetric because is symmetric. The
λ
∈ R
set Ω is thus symmetric. If (2.1) is a finite relative presentation of G with respect
to H , then Ω is a finite set.
λ
{ }
We obtain the following lemma by the same proof as [11, Lemma 2.27].
Lemma 2.6. Let G be a group having the finite relative presentation (2.1) with
respect to H . Let p be a cycle in Γ(G,X ) and for each k =1,2,...,m and
λ
{ } (cid:116)H
λ Λ, let p be an isolated H -component of p. We put M = max R
k ∈ k λk { (cid:107) (cid:107) |
R . Then for any k = 1,2,...,m, we obtain that φ(p ) Ω and
∈ R } k ∈ (cid:104) λk(cid:105)
m φ(p ) M Arearel(φ(p)).
k=1| k |Ωλk ≤ ·
(cid:80)Let p be a path in Γ(G,X ). If the length of any -component of p is equal
(cid:116)H H
to 1, then p is said to be locally minimal. If any -component of p is isolated, we
H
say that p is a path without backtracking. Note that any geodesic in Γ(G,X )
(cid:116)H
is a locally minimal path without backtracking.
In order to define a minimal lift of a locally minimal path without backtracking
in Γ(G,X ), we show the following lemma (see also [14, Lemma 3.2]), which is
(cid:116)H
a generalization of [11, Proposition 2.9].
Lemma 2.7. SupposethatGhasthefiniterelativepresentation(2.1)withrespect
to H and that X is also a finite relative generating set of G with respect to
λ
{ }
K . Then for any λ Λ, the group H is generated by Ω , i.e., H is
λ,µ λ λ λ λ
{ } ∈ (cid:116)K
finitely generated by Ω relative to K .
λ λ,µ µ
{ }
Proof. Note that for each λ Λ, the set Ω is finite.
λ
∈
Wedefineamapπ fromΓ(G,X )toΓ(G,X )asfollows: π istheidentity
(cid:116)K (cid:116)H
on the set G of vertices and on the set of edges labeled by elements of X. For any
λ Λ, any µ M and any edge e labeled by an element of , by using of the
λ λ,µ
∈ ∈ K
inclusion from K into H , π(e) is an edge labeled by an element of .
λ,µ λ λ
H
For each λ Λ, let h be an element of H 1 . We take a geodesic r from 1 to
λ
∈ \{ }
h in Γ(G,X ) and put q =π(r) which is a path in Γ(G,X ). Since 1 and h
(cid:116)K (cid:116)H
are elements of H , we have an edge p in Γ(G,X ) from 1 to h with φ(p) = h
λ
(cid:116)H
(see Figure 1).
If p is equal to q, then φ(r) is an element of by the definition of π. We then
λ
K
obtain φ(p)=φ(r) .
λ
∈(cid:104)K (cid:105)
If p is not equal to q, then we consider a cycle qp 1 in Γ(G,X ). If p 1 is an
− −
(cid:116)H
isolated H -component of qp 1, then h=φ(p) Ω by Lemma 2.6. If p 1 is an
λ − λ −
∈(cid:104) (cid:105)
H -component of qp 1 but not isolated in qp 1, then we denote by q ,q ,...,q
λ − − 1 2 m
all the H -components of q connected to p 1 such that they are arranged on q in
λ −
thisorder. Whenp 1 isnotevenanH -componentofqp 1,letq ,q ,...,q beall
− λ − 1 2 m
the H -components of q which are mutually connected, placed on q in this order,
λ
and 1=(q ) =p or h=(q ) =p holds.
1 m + +
− −
For each k =1,2,...,m, we note φ(q ) . For each k =2,3,...,m, let r
k λ k
∈(cid:104)K (cid:105)
be a subpath of q from (q ) to (q ) . When (q ) = p (resp. (q ) = p ),
k 1 + k 1 m + +
− − − (cid:54) − (cid:54)
let r (resp. r ) be a subpath of q from p to (q ) (resp. (q ) to p ). If
1 m+1 1 m + +
− −
(q ) = p (resp. (q ) = p ), then we regard as r = (resp. r = ). The
1 m + + 1 m+1
− − ∅ ∅
path q in Γ(G,X ) is thereby represented by r q r q q r .
1 1 2 2 m m+1
(cid:116)H ···
In Γ(G,X ), for each k =2,3,...,m, let c be a connector from (q ) to
k k 1 +
(cid:116)H −
(q ) . When(q ) =p ,letc beaconnectorfromp to(q ) inΓ(G,X ). If
k 1 1 1
− − (cid:54) − − − (cid:116)H
(q ) =p , then we regard as c = . When (q ) =p , let c be a connector
1 1 m + + m+1
− − ∅ (cid:54)
8 Y.MATSUDA,S.OGUNI,ANDS.YAMAGATA
q
q
2
r
2
q1 c2 qm
r1 rm+1
c c
1 m+1
1 p h
Figure 1. A cycle qp 1 in Γ(G,X )
−
(cid:116)H
from (q ) to p in Γ(G,X ). If (q ) = p , then we regard as c = .
m + + m + + m+1
(cid:116)H ∅
We thereby obtain cycles c1q1c2q2···qmcm+1p−1 and ckrk−1 in Γ(G,X (cid:116) H) for
each k = 1,2,...,m+1. Since for each k = 1,2,...,m+1, the H -component
λ
ck is isolated in ckrk−1, we obtain φ(ck) ∈ (cid:104)Ωλ(cid:105) by Lemma 2.6. Because of h =
φ(p) = φ(c q c q q c ) = φ(c ) φ(q ) φ(c ) φ(q ) φ(q ) φ(c ), we
1 1 2 2 m m+1 1 1 2 2 m m+1
obtain h Ω ···. ··· (cid:3)
λ λ
∈(cid:104) (cid:116)K (cid:105)
In the setting of Lemma 2.7, we take a locally minimal path p without back-
tracking in Γ(G,X ). Note that Ω is a finite set and X Ω is also a finite
(cid:116)H (cid:116)
relative generating set of G with respect to K . For each λ Λ, by replacing
λ,µ
{ } ∈
every H -component r of p with a geodesic in Γ(H ,Ω ) from r to r , we
λ λ λ λ +
(cid:116)K −
obtain a path p in Γ(G,X Ω ) from p to p . The new path p is called a
+
minimal lift of p. We note t(cid:116)hat e(cid:116)acKh vertex o−f p is also one of the vertices of p and
the inequality l(p) l(p) holds.
(cid:98) (cid:98)
≤
Take any vertices g and g of p and suppose that g and g are arranged on p
1 2 1 2 (cid:98)
(cid:92)
in this order. We denot(cid:98)e by [g1,g2] the subpath of p from g1 to g2.
Let A and B be metric spaces.(cid:98)Take two constants α 1 and β 0. A map(cid:98)
≥ ≥
f: A B is called an (α,β)-quasi-isometric embedding if for any a ,a A, the
(cid:98) 1 2
→ ∈
inequality
1
d(a ,a ) β d(f(a ),f(a )) αd(a ,a )+β
1 2 1 2 1 2
α − ≤ ≤
holds. When A is an interval of R, we call f (and also the image of f) an (α,β)-
quasigeodesic in B. Let f be an (α,β)-quasi-isometric embedding from A into B.
We call f an (α,β)-quasi-isometry if there exists a constant k 0 satisfying that
≥
for each b B, there exists a A with d(f(a),b) k. A map f: A B is simply
∈ ∈ ≤ →
called a quasi-isometric embedding (resp. quasi-isometry) if there exist constants
α 1andβ 0suchthatthemapf isan(α,β)-quasi-isometricembedding(resp.
≥ ≥
(α,β)-quasi-isometry).
Thefollowingisakeylemmainthispaper. Thisisageneralizationof[8,Lemma
4.4] (see another generalization [14, Proposition 3.9]).
ON RELATIVE HYPERBOLICITY FOR A GROUP 9
Lemma 2.8. SupposethatGhasthefiniterelativepresentation(2.1)withrespect
to H and G is hyperbolic relative to H . Suppose that X is also a finite
λ λ
{ } { }
relativegeneratingsetofGwithrespectto K . Thenforanyα 1andβ 0,
λ,µ
{ } ≥ ≥
there exist constants C 1 and D 0 satisfying the following: For any locally
≥ ≥
minimal (α,β)-quasigeodesic p without backtracking in Γ(G,X ), a minimal
(cid:116)H
lift p of p is also a locally minimal (C,D)-quasigeodesic without backtracking in
Γ(G,X Ω ).
(cid:116) (cid:116)K
(cid:98)
Proof. Let δ be the relative Dehn function with respect to (2.1) and let B be a
constant with δ(n) Bn for any n N. We put M =max R R .
≤ ∈ {(cid:107) (cid:107)| ∈R}
Let g be an arbitrary element of G. We denote by p a locally minimal (α,β)-
quasigeodesicfrom1tog withoutbacktrackinginΓ(G,X ). Letpbeaminimal
(cid:116)H
lift of p in Γ(G,X Ω ). We put p=e e e , where for each k =1,2,...,b,
1 2 b
(cid:116) (cid:116)K ···
e is an edge of Γ(G,X Ω ). Set v = 1 = (e ) ,v = (e ) ,...,v =
k 1 1 2 (cid:98)2 b
(cid:116) (cid:116) K − −
(e ) ,v = (e ) = g. We note that the set of vertices of p is contained in that
b b+1 b + (cid:98)
−
of p and p is a locally minimal path without backtracking. Put γ = 2α+β +2,
C =MB(α+3)+α+1 and D =2C+γ(MB+1).
Letustakearbitraryi,j =1,2,...,b+1withi<j andputv =g andv =g .
(cid:98) (cid:98) i 1 j 2
(cid:92)
Claim 2.9. The inequality l([g ,g ]) Cd (g ,g )+γ(MB+1) holds.
1 2 X Ω 1 2
≤ (cid:116) (cid:116)K
Proof. If for some λ Λ, both g and g lie in a minimal lift of an H -component
1 2 λ
∈
of p, then [g ,g ] denotes an edge from g to g in Γ(G,X ) labeled by an
1 2 1 2
(cid:116) H
element of . Otherwise, [g ,g ] denotes a path from g to g in Γ(G,X ) as
λ 1 2 1 2
H (cid:116)H
follows: Let k (resp. k ) be the minimal (resp. maximal) number of 1,2,...,b+1
1 2
(cid:92) (cid:92)
with v p [g ,g ] (resp. v p [g ,g ]). We put v = g and v = g .
k1 ∈ ∩ 1 2 k2 ∈ ∩ 1 2 k1 3 k2 4
When g (resp. g ) is a vertex of p, the equality g = g (resp. g = g ) holds.
1 2 1 3 2 4
We denote by [g ,g ] (resp. [g ,g ]) an edge from g to g (resp. from g to g ) in
1 3 4 2 1 3 4 2
Γ(G,X ) labeled by an element of because for some λ Λ, both g and g
λ 1 3
(cid:116)H H ∈
(reap. g and g ) lie in a minimal lift of an H -component of p. If g = g (resp.
4 2 λ 1 3
g =g ), then we regard as [g ,g ]= (resp. [g ,g ]= ). The subpath of p from
2 4 1 3 4 2
∅ ∅
g to g is denoted by [g ,g ]. If k = k , then we also regard as [g ,g ] = . Put
3 4 3 4 1 2 3 4
∅
[g ,g ]=[g ,g ][g ,g ][g ,g ]. This is a path in Γ(G,X ).
1 2 1 3 3 4 4 2
(cid:116)H
Whether both g and g lies in a minimal lift of an -component of p or not,
1 2
H
the path [g ,g ] is a locally minimal (α,γ)-quasigeodesic without backtracking in
1 2
(cid:92)
Γ(G,X ) by [11, Lemma 3.5] and [g ,g ] is a minimal lift of [g ,g ].
1 2 1 2
(cid:116)H
Let r be a geodesic from g to g in Γ(G,X Ω ). Since for each λ Λ and
(cid:48) 1 2
(cid:116) (cid:116)K ∈
µ M , the group H contains K as a subgroup, the following inequality holds;
λ λ λ,µ
∈
l([g ,g ]) αd (g ,g )+γ
1 2 X 1 2
≤ (cid:116)H
αd (g ,g )+γ =αl(r )+γ.
X Ω 1 2 (cid:48)
≤ (cid:116) (cid:116)K
Let us regard r as a path r in Γ(G,X ). Note that l(r)=l(r ). If r is equal
(cid:48) (cid:48)
(cid:116)H
to [g ,g ], then r is a minimal lift of [g ,g ]. We then have
1 2 (cid:48) 1 2
(cid:92)
l([g ,g ])=l(r )=d (g ,g ).
1 2 (cid:48) X Ω 1 2
(cid:116) (cid:116)K
(cid:92) (cid:92)
If r = [g ,g ], then l([g ,g ]) = l(r ) = d (g ,g ). In the following, we
(cid:48) 1 2 1 2 (cid:48) X Ω 1 2
(cid:92) (cid:116) (cid:116)K
assume that r =[g ,g ] and r =[g ,g ].
1 2 (cid:48) 1 2
(cid:54) (cid:54)
10 Y.MATSUDA,S.OGUNI,ANDS.YAMAGATA
I
1,i1
I
1,3
I1,2 c1,2
c
1,i1
c
1,1
I
1,1
c
1,0 a
1
g1 h1 g2
Figure 2. Connectors and a cycle a
1
We treat two cases where (i) each -component of [g ,g ] 1 is also an -
1 2 −
H H
component of r[g ,g ] 1 and isolated in r[g ,g ] 1 and (ii) there exists an -
1 2 − 1 2 −
H
component of [g ,g ] 1 such that it is also an -component of r[g ,g ] 1 but not
1 2 − 1 2 −
H
isolated in r[g ,g ] 1, or it is not even an -component of r[g ,g ] 1.
1 2 − 1 2 −
H
(i) Let h ,h ,...,h be all the -components of [g ,g ] and for each k =
1 2 n 1 2
H
1,2,...,n, let h be a minimal lift of h in Γ(G,X Ω ). By Lemma 2.6,
k k
(cid:116) (cid:116) K
we obtain the inequality
n (cid:99) n
l(h ) φ(h ) M Arearel(φ(r[g ,g ] 1))
k k Ω 1 2 −
≤ | | ≤ ·
k=1 k=1
(cid:88) (cid:88)
(cid:99) M δ(l(r[g ,g ] 1)) MB l(r[g ,g ] 1)
1 2 − 1 2 −
≤ · ≤ ·
MB l([g ,g ] 1)+l(r) MB αl(r )+γ+l(r )
1 2 − (cid:48) (cid:48)
≤ { }≤ { }
=MB(α+1) l(r )+MBγ.
(cid:48)
·
By the above inequality, the following inequality holds;
n
(cid:92)
l([g ,g ]) l(h )+l([g ,g ])
1 2 k 1 2
≤
k=1
(cid:88)
MB(α(cid:99)+1) l(r )+MBγ+αl(r )+γ
(cid:48) (cid:48)
≤ ·
= MB(α+1)+α l(r )+γ(MB+1)
(cid:48)
{ }
= MB(α+1)+α d (g ,g )+γ(MB+1)
X Ω 1 2
{ } (cid:116) (cid:116)K
Cd (g ,g )+γ(MB+1).
X Ω 1 2
≤ (cid:116) (cid:116)K
(ii) Let h1 be one of the H-components of [g1,g2] such that h−11 is also an
H-component of r[g1,g2]−1 but not isolated in r[g1,g2]−1, or h−11 is not even an
-component of r[g ,g ] 1. We assume that on the subpath of [g ,g ] from g to
1 2 − 1 2 1
H
(h ) , there is no such -component of [g ,g ]. Since [g ,g ] is a path without
1 1 2 1 2
− H
backtracking in Γ(G,X ), it does not contain any -components which are
(cid:116) H H
connected to h1. We assume that h1 is an Hλ1-component of [g1,g2]. If h−11 is
an H -component of r[g ,g ] 1 but not isolated in r[g ,g ] 1, then the path r in
λ1 1 2 − 1 2 −
Γ(G,X ) contains all the edges of r[g ,g ] 1 labeled by elements of which
(cid:116)H 1 2 − Hλ1