Quasisymmetric maps of boundaries of amenable hyperbolic groups

18 Pages·2012·0.27 MB·English

Quasisymmetric maps of boundaries of
amenable hyperbolic groups
Tullia Dymarz ∗
July 31, 2012
Abstract
InthispaperweshowthatifY = N×Q isametricspacewhereN
m
is a connected, simply connected, nilpotent Lie group endowed with
an admissible metric then any quasisymmetic map of Y is actually
bilipschitz. A metric on N is admissible if it makes Y into a parabolic
visual boundary of a mixed type locally compact amenable hyperbolic
group. We also prove some rigidity results on uniform subgroups of
bilipschitz maps of Y in the case where N = Rn.
MathematicsSubjectClassication(2000). 20F65,30C65,53C20.
Keywords. quasi-isometry,quasisymmetricmap,negativecurvature.
1 Introduction
Quasisymmetric maps were introduced in [BA56] as natural replacements for
quasiconformal maps for metric spaces where classical quasiconformal maps
do not make sense (see Section 2 for the definition). For many standard
metric spaces (such as Rn) quasisymmetric maps are much more abundant
than bilipschitz maps. For Y = N ×Q this is not the case.
m
Theorem 1 Let N be a nilpotent Lie group with an admissible metric d and
Q the m-adics with the standard metric. Then any quasisymmetric map of
m
Y = N ×Q onto itself is bilipschitz.
n
∗Partially supported by NSF grant 1207296. Deptarment of Mathematics, University
of Wisconsin, Madison, 480 Lincoln Dr. 53706. email: [email protected]
1
Quasisymmetricmapsareespeciallyinterestingsincetheyarestronglylinked
to negative curvature geometry in that quasi-isometries of negatively curved
spaces induce quasisymmetric maps of their visual boundaries. We are able
to study Y precisely because it is the parabolic visual boundary of a geodesic
negatively curved space X that fibers over a simplicial m+1 valent tree
N,ϕ,m
T . Quasisymmetric maps of Y = N × Q = ∂X are induced by
m+1 m N,ϕ,m
quasi-isometries of X while bilipschitz maps of Y are induced by height-
N,ϕ,m
respecting quasi-isometries (see Section 3 for the definition). Using coarse
topology provided by [FM00] we prove Theorem 1 by showing that all quasi-
isometries of X are height-respecting.
N,ϕ,m
This result should be compared to results of Xie in [SX, Xiec, Xiea, Xieb]
which use the same dictionary but in the reverse direction to show that all
quasi-isometries of certain negatively curved homogeneous spaces are height-
respecting by proving directly that all quasisymmetric maps of their bound-
aries are bilipschitz.
In fact, Theorem 1 can be thought of as an extension of Xie’s results. In
[Hei74], Heintze showed that all negatively curved homogeneous spaces can
be given as solvable Lie groups (see Section 3.1 for details). In a similar spirit
[CdCMT], Caprace et al. classify all locally compact amenable hyperbolic
groups. They show that there are three types of non-elementary amenable
hyperbolic locally compact groups: negatively curved homogeneous spaces,
stabilizers of an end in the full automorphism group of a semi-regular locally
finite tree, and combinations of the two via a warped product construction.
It is this last type that we call mixed type. All mixed type amenable hyper-
bolic groups act properly on some X .
N,ϕ,m
Remark. Theorem 1 should hold in more generality. For example it should
hold for any space whose hyperbolic cone (see [BS00]) satisfies the conditions
imposedinTheorems7.3and7.7in[FM00]. (SeeSection4forthestatements
of these theorems.)
1.1 Other results.
Using the same arguments as Corollary 1.3 in [SX] we can also show the
following:
Corollary 2 Therearenofinitelygeneratedgroupsquasi-isometrictoX .
N,ϕ,m
2
Furthermore, from the appendix of [FM98] we know that Q and Q are
m m(cid:48)
bilipschitz equivalent only if m = ri and m = rj for some common base r.
This gives us a partial quasi-isometry classification result.
Corollary 3 X is not quasi-isometric to X if m,m(cid:48) are not pow-
N,ϕ,m N(cid:48),ϕ(cid:48),m(cid:48)
ers of a common base.
A more detailed classification result can be deduced in many cases from
[FM00, Pen11, Ahl02] but we will not give the details here except in case of
N = Rn. Note that if we specialize to N = Rn then ϕ = ϕ is given by
M
multiplication by an n×n matrix M whose eigenvalues all have norm greater
than one. Using work of [FM00] we get a full quasi-isometry classification in
this case.
Corollary 4 X is quasi-isometric to X if and only if m = ri,
Rn,ϕM,m Rn,ϕM(cid:48),m(cid:48)
m(cid:48) = rj and M and M(cid:48) have absolute Jordan forms that are powers of each
other.
We also extend results from [Dym10] and prove that certain groups of bilip-
schitz maps of Rn×Q can be conjugated to be of a particularly nice form.
m
We state the theorem here but all definitions are given in the appendix.
Theorem 5 Let U be a uniform separable subgroup of Bilip (Rn×Q ) that
M m
acts cocompactly on the space of distinct pairs of points of Rn ×Q . Then
m
U can be conjugated into ASim (Rn ×Q ) for some p.
M p
TheprooffollowsTheorem2in[Dym10]verycloselywhichiswhywerelegate
this result to the appendix and provide only a brief outline.
1.2 Outline
Following some preliminaries in Section 2 we study the geometry and bound-
aries of X in Section 3. We prove Theorem 1 in Section 4. In the
N,ϕ,m
appendix we prove Theorem 5.
Acknowledgements. The author would like to thank Xiangdong Xie for
usefulconversationsandbothXiangdongXieandJohnMackayforcomments
on an earlier draft.
3
2 Preliminaries
Definition 1 A map f : X → Y between metric spaces is called a
Quasisymmetric embedding if for some homeomorphism η : [0,∞) →
[0,∞)
(cid:18) (cid:19)
d (f(y),f(x)) d (y,x)
Y X
≤ η .
d (f(y),f(x(cid:48))) d (y,x(cid:48))
Y X
Bilipschitz embedding if
a d(x,y) ≤ d(f(x),f(y)) ≤ b d(x,y).
If we can chose a = 1/K and b = K then we say f is K-bilipschitz. If
we can chose a = s/K and b = sK then we say that f is a (K,s)-quasi-
similarity. We say that a group of bilipschitz maps/quasi-similarities
is uniform if K is uniform over all group elements.
(K,C)-Quasi-isometric embedding if
1
−C + d (x,x(cid:48)) ≤ d (f(x),f(x(cid:48))) ≤ Kd (x,x(cid:48))+C
X Y X
K
S-Similarity
d (f(x),f(x(cid:48))) = Sd (x,x(cid:48))
Y X
Uniform embedding if for some ρ : [0,∞) → [0,∞) with ρ(t) → ∞ as
t → ∞
ρ(d (x,x(cid:48))) ≤ d (f(x),f(x(cid:48))) ≤ Kd (x,x(cid:48))+C
X Y X
3 CAT(−k) spaces and their visual boundaries
Let X be a CAT(−k) metric space. Riemannian manifolds with sectional
curvature≤ −k andtheirconvexsubsetsaretheprimeexamplesofCAT(−k)
spaces. Gluing two CAT(−k) spaces along closed convex subsets also results
in a CAT(−k) subspace. Note that up to rescaling the metric by a constant
we can assume that k = 1.
Given a CAT(−1) space X and a ∈ ∂X and we define the Euclid-Cygan
metric on ∂X − {a} as is done in the appendix in [HP97]. Let H be a
4
horosphere centered at a and b,c ∈ ∂X − {a}. Let b ,c be two geodesics
t t
connectingb,ctoawithb ,c ∈ H. (Notethatthisorientationistheopposite
0 0
of what is given in [HP97]). The Euclid-Cygan metric on ∂X −{a} is given
by
da,H(b,c) = lim e12(2t+dX(bt,ct)).
t→−∞
We can also endow ∂X −{a} with a visual parabolic metric. For any b,c ∈
∂X −{a} the visual parabolic metric is given by
d¯ (b,c) = et0
a,H
where t is the point at which d (b ,c ) = 1. It is easy to see that the two
0 X t0 t0
metrics are bilipschitz equivalent since if t ≤ t then
0
2(t −t)+1−C ≤ d (b ,c ) ≤ 2(t −t)+1+C
0 X t t 0
and so
da,H(b,c) = lim e12(2t+dX(bt,ct))
t→−∞
(cid:39) lim e12(2t+2(t0−t)+1)
t→−∞
(cid:39) et0 = d¯ (b,c).
a,H
(See [SX] for more details).
Remarks. Note that the reason e is chosen as a base here is because X is
CAT(−1). If X were CAT(−k) the base would be e√1k instead of e. We
can in fact choose the base to be eα for any α < 1. This is equivalent to
snowflaking the metric.
Definition 2 For any b ∈ ∂X−{a} there is a unique geodesic b connecting
t
b to a with b ∈ H. We call such geodesic a vertical geodesic.
0
In this paper we will only be working with CAT(−k) spaces that have the
property that for every x ∈ X there is a vertical geodesic passing through x.
In this case we can view X as a union (not necessarily disjoint) of all vertical
geodesics
(cid:91)
X = b .
t
b∈∂X−{a}
5
Definition 3 We endow X with a height function (this is just a horofunc-
tion) h : X → R given by h(x) = t if x = b . Note that if x = b and x = b(cid:48)
t t t(cid:48)
then necessarily t = t(cid:48).
A quasi-isometry of X induces a quasisymmetric map
∂f : ∂X −{a} → ∂X −{f(a)}
¯
with respect to either d or d . Since the two metrics are bilipschitz
a,H a,H
equivalent we will drop the distinction between them and simply write d .
∞
In the following sections we describe all of the CAT(−1) spaces we will be
working with.
3.1 Negatively curved homogeneous spaces
Let N be a connected, simply connected, nilpotent Lie group. We say that a
one parameter group of automorphisms ϕ is contracting (resp. expanding)
t
if for all g ∈ N we have ϕ (g) → 1 (resp. ϕ (g) → 1) as t → ∞.
t −t
If ϕ : N → N is a one parameter group of expanding automorphisms
t
then G = N (cid:111) R is a negatively curved homogeneous space when endowed
ϕ ϕ
with an appropriate left invariant Riemannian metric [Hei74, CdCMT]. By
[Hei74] we have that, up to rescaling the metric, all N (cid:111) R can be endowed
ϕ
with a CAT(−1) metric. The geometry of these spaces have been studied
by [FM00, Pen11, Ahl02]. Some of their boundaries have been analyzed in
[Dym10, DP11, SX, Xiec, Xiea, Xieb].
Since G is negatively curved, we can consider its parabolic visual bound-
ϕ
ary ∂G − {∞} where ∞ is chosen so that vertical geodesics are be given
ϕ
by
γ (t) = (g,t) ∈ N (cid:111) R.
g ϕ
We can identify ∂G −{∞} with N and then the height function is simply
ϕ
given by h(g,t) = t.
To get a better grasp of the metric G we define a quasi-isometrically
ϕ
equivalent metric on G that is easier to work with than a Riemannian
ϕ
metric. Let d be a metric induced by a left invariant metric on N and
N
for each t ∈ R set
d (g ,g ) = d (ϕ (g ),ϕ (g )) = (cid:107)ϕ (g−1g )(cid:107).
t 1 2 N −t 1 −t 2 −t 1 2
6
¯
Let d be the largest metric on G such that the vertical geodesics given above
ϕ
areactuallygeodesicsandthedistanceoneachlevelseth−1(t) = N×{t} (cid:39) N
is d [Gro87]. Note also that the level set N × {t} with the metric d is
t t
exponentially distorted in G . With this metric it is easy to verify that for
ϕ
two distinct vertical geodesics γ and γ we have
g0 g1
lim d(γ (−t),γ (−t)) = ∞ and lim d(γ (t),γ (t)) = 0.
t→∞ g0 g1 t→∞ g0 g1
Note that if d (g ,g ) = 1 then 1/K −C ≤ d((g ,t ),(g ,t )) ≤ K +C so
t0 0 1 0 0 1 0
that up to bilipschitz equivalence we can interpret the parabolic visual metric
as
d (g ,g ) = et0
∞ 0 1
where t is the smallest value at which d (g ,g ) = 1. See Section 5 of [SX]
0 t0 0 1
for more details.
3.1.1 Snowflaking
It is worthwhile to note that by reparametrizing ϕ as ϕ(cid:48) = ϕ we get
t t αt
boundary metrics on G and G that are snowflake equivalent. In particular
ϕ ϕ(cid:48)
d = dα .
∞,ϕ ∞,ϕ(cid:48)
Note that there is always a range of admissible α that ensure that dα is
∞,ϕ(cid:48)
actually a metric. Alternatively we could have chose the base a = eα instead
of e in our definition of boundary metric. This will be important later when
we define the millefeuille space (see Section 3.3).
3.2 Trees
Any tree T is a CAT(−1) space so again by fixing a point ξ at infinity we
can define the parabolic visual boundary with respect to ξ. Picking a point
at infinity induces an orientation on edges (towards the point at infinity).
This in turn induces a height function: designate a base point vertex to be
at height zero then use orientation to determine the heights of all of the other
vertices. Vertical geodesics in T are the geodesics that are compatible with
the height function. The parabolic visual boundary is again just the set of
verticalgeodesicsandinthiscasetheEuclid-Cyganmetriccanbeinterpreted
as the et0 where t is the height at which the two vertical geodesics first
0
coincide. Note that for a tree we can define a parabolic visual metric at0 for
any base a > 1 and still have it be a metric.
7
3.3 Millefeuille space
Let T be the regular m + 1 valent tree with orientation such that each
m+1
vertex has m incoming edges and one outgoing edge. With this orientation
there is a natural choice for ∞ ∈ ∂T . Again, vertical geodesics are given
m+1
by the coherently oriented infinite geodesics. In this case we can identify
∂T −{∞} with the m-adics Q (see [FM98] for the identification).
m+1 m
Definition 4 (Millefeuille space) Let G = N(cid:111) R be a negatively curved
ϕ ϕ
homogeneous space with height function h : G → R. Let h : T → R
ϕ ϕ m m+1
be a height function. The millefeuille space is defined to be
X = {(x,y) ∈ G ×T | h (x) = h (y)}
N,ϕ,m ϕ m+1 ϕ m
with the induced L∞ metric.
Alternatively we can view X as the metric fibration
N,ϕ,m
π : X → T
N,ϕ,m m+1
where π−1((cid:96)) is identified with G for each coherently oriented line (cid:96) in T
ϕ m+1
via a height-preserving isometry.
This space X was first defined in Section 7 of [CdCMT]. In that
N,ϕ,m
section, it is also noted that X is a CAT(−k) space if G is CAT(−k).
N,ϕ,m ϕ
This is because locally X is obtained by glueing m copies of G , along
N,ϕ,m ϕ
closed convex subsets (namely along horoballs of G ).
ϕ
Definition 5 Following the terminology coined by Farb-Mosher in [FM00],
we call each π−1((cid:96)) a hyperplane of X and each π−1(v) a horizontal leaf.
As with G and T there is a natural choice of ∞ ∈ X such that
ϕ m+1 N,ϕ,m
the vertical geodesics in X are precisely the geodesics that project to
N,ϕ,m
vertical geodesics in both G and T . There is also an obvious induced
ϕ m+1
height function h : X → R given by h(x,y) = h (x) = h (y).
N,ϕ,m ϕ m
Proposition 6 The parabolic visual boundary ∂X −{∞} is bilipschitz
N,ϕ,m
equivalent to (N ×Q ,d ) where d is the maximum of the metrics d
m ϕ,m ϕ,m ϕ
(on N) and d (on Q ).
Qm m
8
Proof. Note that for any two distinct vertical geodesics γ and γ(cid:48) we have
three possible cases. If γ and γ(cid:48) project to the same geodesic in T then
m+1
γ,γ(cid:48) both lie in the same hyperplane f−1((cid:96)) (cid:39) G in which case we can
ϕ
identify γ,γ(cid:48) with vertical geodesics in G (namely γ (cid:39) γ and γ(cid:48) (cid:39) γ for
ϕ g0 g1
some g ,g ∈ N). Then
0 1
d (γ,γ(cid:48)) = d (g ,g ).
∞,X ∞,Gϕ 0 1
Likewise,iftheprojectionofγ andγ(cid:48) toG isthesamethenthetwogeodesics
ϕ
coincide above some t ∈ Z. In this case γ and γ(cid:48) lie in the same copy of
0
T and so we have
m+1
d (γ,γ(cid:48)) = d (γ,γ(cid:48)).
∞,X ∞,Tm+1
Finallythelastcaseiswhenγ andγ(cid:48) projecttotwodifferentverticalgeodesics
inbothfactors. Nevertheless,eventually(aboveheightt ),thesetwogeodesics
1
lie in the same hyperplane f−1((cid:96)) (cid:39) G . Then, above t , we can identify
ϕ 1
γ (cid:39) γ and γ(cid:48) (cid:39) γ for some g ,g ∈ N. If d (γ(t ),γ(cid:48)(t )) ≥ 1 then
g0 g1 0 1 t1 1 1
d (γ,γ(cid:48)) = d (g ,g )
∞,X ∞,Gϕ 0 1
since then the height at which γ,γ(cid:48) are distance one is above t . Otherwise,
1
d (γ(t ),γ(cid:48)(t )) < 1 and the boundary metric has the property that
t1 1 1
1/K d (γ,γ(cid:48)) ≤ d (γ,γ(cid:48)) ≤ K d (γ,γ(cid:48)).
∞,Tm+1 ∞,X ∞,Tm+1
Finally, in order to get the standard metric on Q (i.e. d (γ,γ(cid:48)) =
m ∞,Tm+1
mt0 where t is the height at which γ,γ(cid:48) initially come together) we must
0
snowflake our boundary metric by α = lnm. To ensure that is possible we
might have to replace ϕ(t) with ϕ(cid:48)(t) = ϕ(αt). (See the comments at the
end of Section 3.1).
3.4 Quasi-isometries
Definition 6 Let X be a CAT(−1) space with height function h : X → R.
We say that a quasi-isometry f : X → X is height-respecting if there is a
constant A such that f maps any height level set of h to within distance A of
a height level set and if the map induced on height is bounded distance from
a translation. In other words there exists a constant a such that if h(x) = t
then
−C +t+a ≤ h(f(x)) ≤ t+a+C.
9
It is now a well known fact (Lemma 6.1 [Xiea] and [FM98]) that for G
ϕ
and T height-respecting quasi-isometries (up to bounded distance) are
m+1
in one-to-one correspondence with bilipschitz maps of the parabolic visual
boundary. Combining these two facts we can see that the same is true for
X .
N,ϕ,m
Proposition 7 Any height-respecting quasi-isometry of X induces a
N,ϕ,m
bilipschitz maps of the parabolic visual boundary
∂X −{∞} (cid:39) N ×Q .
N,ϕ,m m
Remark. Note that while we prove here that all quasi-isometries of X
N,ϕ,m
are height-respecting it is not always the case that all quasi-isometries of
T and G are height-respecting. This is clear for T however for G
m+1 ϕ m+1 ϕ
the answer is more subtle. For instance when ϕ (x) = etx the space Rn (cid:110)
t ϕ
R is isometric to Hn+1 whose quasi-isometries can be identified with the
quasiconformal maps of Sn. The other rank one symmetric spaces can also
be written as N (cid:111) R for the appropriate N and ϕ. In [SX, Xiec, Xiea], Xie
ϕ
showed that when Rn(cid:111) R is not isometric to Hn+1 then all quasi-isometries
ϕ
are height-respecting. In [Xieb], he was able to show that the same result is
true for certain N(cid:111) R. It is an open question whether all quasi-isometries of
ϕ
negatively curved homogeneous spaces that are not isometric to symmetric
spaces are height-respecting.
4 Proof of Theorem 1
Inthissectionweshowthatanyquasi-isometryofX isheight-respecting.
N,ϕ,m
We start with Farb-Mosher’s Theorem 7.7 [FM00].
Theorem 8 (Theorem 7.7 in [FM00]) Let π : X → T be a metric fibra-
tion over a bushy tree T such that the fibers of π are contractible k-manifolds
for some k. Let f : X → X be a self quasi-isometry. Then there exists a con-
stant A, depending only on the metric fibration data of π, the quasi-isometry
data of f and T such that
1. For each hyperplane P ⊂ X there exists a unique hyperplane Q ⊂ X
such that d (f(P),Q) ≤ A.
H
10