8 Analytical aspects of boundaries of groups

We begin with the basic definitions of the quasiconformal analysis. For a quadruple of points 

dvipng error! exitcode was 2 (signal 0), transscript follows:
in a metric space 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, 
dvipng error! exitcode was 2 (signal 0), transscript follows:
denotes their cross-ratio, i.e. 

dvipng error! exitcode was 2 (signal 0), transscript follows:

Quasiconformal (analytic definition): A homeomorphism 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is quasiconformal iff 

  1. dvipng error! exitcode was 2 (signal 0), transscript follows:
    and

  2. There exists 

    dvipng error! exitcode was 2 (signal 0), transscript follows:
    so that 
    dvipng error! exitcode was 2 (signal 0), transscript follows:
    a.e. (here 
    dvipng error! exitcode was 2 (signal 0), transscript follows:
    is the Jacobian of 
    dvipng error! exitcode was 2 (signal 0), transscript follows:
    ).

The essential supremum 

dvipng error! exitcode was 2 (signal 0), transscript follows:
of 
dvipng error! exitcode was 2 (signal 0), transscript follows:
on 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is called the coefficient of quasiconformality of 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. A mapping 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is called 
dvipng error! exitcode was 2 (signal 0), transscript follows:
-quasiconformal if 
dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Note: The map 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is differentiable almost everywhere, so the derivative and Jacobian in (2) make sense pointwise a.e.. The assumption (1) can be replaced by the assumption that 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is ACL, i.e., that 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is absolutely continuous on a.e.\ line parallel to the 
dvipng error! exitcode was 2 (signal 0), transscript follows:
coordinate directions. The above analytical definition of quasiconformality for maps of 
dvipng error! exitcode was 2 (signal 0), transscript follows:
turns out to be equivalent to four other definitions given below.

1. Quasiconformal (metric definition): Let 

dvipng error! exitcode was 2 (signal 0), transscript follows:
be a homeomorphism. For 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, define the following: 

dvipng error! exitcode was 2 (signal 0), transscript follows:



dvipng error! exitcode was 2 (signal 0), transscript follows:



dvipng error! exitcode was 2 (signal 0), transscript follows:

Then, 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is quasiconformal iff 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is uniformly bounded by some 
dvipng error! exitcode was 2 (signal 0), transscript follows:
.

2. Quasiconformal (geometric definition): Let 

dvipng error! exitcode was 2 (signal 0), transscript follows:
be a family of paths in 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. We say a Borel function 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is admissible for 
dvipng error! exitcode was 2 (signal 0), transscript follows:
iff for every 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, we have 
dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Define 

dvipng error! exitcode was 2 (signal 0), transscript follows:
. A homeomorphism 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is quasiconformal iff there is some constant 
dvipng error! exitcode was 2 (signal 0), transscript follows:
such that for every path family 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, we have 

dvipng error! exitcode was 2 (signal 0), transscript follows:

3. A homeomorphism 

dvipng error! exitcode was 2 (signal 0), transscript follows:
between metric spaces 
dvipng error! exitcode was 2 (signal 0), transscript follows:
and 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is quasisymmetric iff there exists a homeomorphism 
dvipng error! exitcode was 2 (signal 0), transscript follows:
such that for all triples of distinct points 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, the following inequality holds: 

dvipng error! exitcode was 2 (signal 0), transscript follows:

4. A homeomorphism 

dvipng error! exitcode was 2 (signal 0), transscript follows:
between metric spaces 
dvipng error! exitcode was 2 (signal 0), transscript follows:
and 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is quasi-Moebius iff there exists a homeomorphism 
dvipng error! exitcode was 2 (signal 0), transscript follows:
such that for all quadruples of distinct points 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, the following inequality holds: 

dvipng error! exitcode was 2 (signal 0), transscript follows:

Note that Definitions 1--4 make sense in the context of general metric spaces, see below for details. If 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is noncompact then quasi-symmetric maps are the same as quasi-Moebius maps. However, for compact metric spaces quasi-Moebius is a more appropriate (although more cumbersome) definition. One can rectify this problem by redefining quasi-symmetric maps for compact metric spaces as follows. A map 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is quasi-symmetric if 
dvipng error! exitcode was 2 (signal 0), transscript follows:
admits a finite covering by open spaces 
dvipng error! exitcode was 2 (signal 0), transscript follows:
so that the restriction 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is quasi-symmetric in the above sense for each 
dvipng error! exitcode was 2 (signal 0), transscript follows:
.

One defines a quasi-symmetric equivalence for metric spaces by 

dvipng error! exitcode was 2 (signal 0), transscript follows:

if there exists a quasisymmetric homeomorphism 

dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Let 

dvipng error! exitcode was 2 (signal 0), transscript follows:
be a metric measure space, where 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is a Borel measure. Then 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is called Ahlfors 
dvipng error! exitcode was 2 (signal 0), transscript follows:
-regular, if there exists 
dvipng error! exitcode was 2 (signal 0), transscript follows:
so that 

dvipng error! exitcode was 2 (signal 0), transscript follows:

for each 

dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Remark Let 

dvipng error! exitcode was 2 (signal 0), transscript follows:
be a metric space with the Hausdorff dimension 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. Then the most natural measure to use is the 
dvipng error! exitcode was 2 (signal 0), transscript follows:
-Hausdorff measure on 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. This is the measure to be used for the boundaries of hyperbolic groups. Then 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is called Ahlfors regular if it is Ahlfors 
dvipng error! exitcode was 2 (signal 0), transscript follows:
-regular with 
dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Given two compact continua 

dvipng error! exitcode was 2 (signal 0), transscript follows:
in a metric space 
dvipng error! exitcode was 2 (signal 0), transscript follows:
define their relative distance 

dvipng error! exitcode was 2 (signal 0), transscript follows:

Here 

dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Given an Ahlfors 

dvipng error! exitcode was 2 (signal 0), transscript follows:
-regular metric measure space 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, define 
dvipng error! exitcode was 2 (signal 0), transscript follows:
to be 

dvipng error! exitcode was 2 (signal 0), transscript follows:

where 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is the set of all curves in 
dvipng error! exitcode was 2 (signal 0), transscript follows:
connecting 
dvipng error! exitcode was 2 (signal 0), transscript follows:
to 
dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Definition 8.1 A metric space 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is called 
dvipng error! exitcode was 2 (signal 0), transscript follows:
- Loewner if it satisfies the inequality 

dvipng error! exitcode was 2 (signal 0), transscript follows:

for a certain function 

dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Problem 8.1 (Comments) [Mario Bonk] Are diffeomorphisms 

dvipng error! exitcode was 2 (signal 0), transscript follows:
dense in the space of all quasiconformal maps?

Remark [Juha Heinonen] The answer is known to be yes" for 

dvipng error! exitcode was 2 (signal 0), transscript follows:
due to Moise's theorem MR0488059.

Remark [Misha Kapovich] In fact, the answer is also known to be "yes" for quasiconformal diffeomorphisms of 

dvipng error! exitcode was 2 (signal 0), transscript follows:
, 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. This was proven by Connell MR0154289 for stable homeomorphisms of 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, improved by Bing MR0238284 to cover dimensions 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. Lastly, it was shown by Kirby MR0242165 that all orientation-preserving homeomorphisms of 
dvipng error! exitcode was 2 (signal 0), transscript follows:
are stable for 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. Note that the proof in the case of quasiconformal homeomorphisms is easier since quasi-conformal homeomorphisms are differentiable a.e. and the stable homeomorphism conjecture was known for 
dvipng error! exitcode was 2 (signal 0), transscript follows:
prior to Kirby's work. However the problem appears to be open in the case 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. On the other hand, Kirby observed that for sufficiently large 
dvipng error! exitcode was 2 (signal 0), transscript follows:
there are open connected subsets 
dvipng error! exitcode was 2 (signal 0), transscript follows:
and a homeomorphism 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, which cannot be approximated by diffeomorphisms 
dvipng error! exitcode was 2 (signal 0), transscript follows:
.

The problem becomes more subtle if we require approximating diffeomorphisms to be globally quasiconformal:

Problem 8.2 (Comments) [Misha Kapovich] Let 

dvipng error! exitcode was 2 (signal 0), transscript follows:
be a quasiconformal homeomorphism. Can 
dvipng error! exitcode was 2 (signal 0), transscript follows:
be approximated by globally quasiconformal diffeomorphisms 
dvipng error! exitcode was 2 (signal 0), transscript follows:
? Can this be done so that 
dvipng error! exitcode was 2 (signal 0), transscript follows:
's are 
dvipng error! exitcode was 2 (signal 0), transscript follows:
-quasiconformal for all 
dvipng error! exitcode was 2 (signal 0), transscript follows:
?

Note that all the maps in problem will extend quasiconformally to the closed 

dvipng error! exitcode was 2 (signal 0), transscript follows:
-ball.

Problem 8.3 (Comments) [Mario Bonk] Find good classes of spaces such that the infinitesimal metric condition (for quasiconformality) implies the local condition. (This is generally true in Loewner spaces.)

Problem 8.4 (Comments) [Kim Ruane] Outside of the boundaries of Fuchsian buildings, what boundaries have the Loewner property?

Remark Loewner spaces are good for analytic tools: have a cotangent bundle, so can "do calculus"; also, can do PDEs, etc.

Problem 8.5 (Comments) [Juha Heinonen] Let 

dvipng error! exitcode was 2 (signal 0), transscript follows:
be a non-smoothable closed simply connected 4-manifold. Does it admit an Ahlfors 4-regular linearly locally contractible metric? This is wide open; unknown even for examples, like 
dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Remark The non-smoothable closed simply-connected 4-manifolds like 

dvipng error! exitcode was 2 (signal 0), transscript follows:
are known not to admit a quasiconformal atlas, MR1032074. In dimensions 
dvipng error! exitcode was 2 (signal 0), transscript follows:
Sullivan MR0537749 proved that every topological manifold admits a quasiconformal atlas and, moreover, quasiconformal structure is unique. An alternative proof of Sullivan's theorem and its generalization was given by J.~Luukkainen in MR1804561, his proof avoids the construction of almost parallelizable hyperbolic manifolds; see also MR0658932. It was observed by Tom Farrell that a detailed proof of the fact that all closed hyperbolic 
dvipng error! exitcode was 2 (signal 0), transscript follows:
-manifolds are virtually almost parallelizable (and much more) is contained in the paper by B.~Okun MR1875614.

Hence in dimension 

dvipng error! exitcode was 2 (signal 0), transscript follows:
one would ask for Ahlfors 
dvipng error! exitcode was 2 (signal 0), transscript follows:
-regular linearly locally contractible metrics on the unresolvable homology manifolds, see MR1394965. The broad goal here is to find an analytic framework for studying exotic topological and homology manifolds.

Problem 8.6 (Comments) [Uri Bader] Develop a theory for analysis on the ideal boundaries of relatively hyperbolic groups, as it is done for hyperbolic groups.

Problem 8.7 (Comments) [Bruce Kleiner] In what generality does quasiconformal imply quasisymmetric? (Specifically, of interest are self-similar spaces which are connected, without local cut points; or visual boundaries of hyperbolic groups.)

Definition. Call a metric space 

dvipng error! exitcode was 2 (signal 0), transscript follows:
quasi-isometrically cohopfian if each quasi-isometric embedding 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is a quasi-isometry. (Examples include Poincar\'e duality groups, solvable groups, Baumslag-Solitar groups.) 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is quasisymmetrically cohopfian if every quasisymmetric embedding 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is onto.

The above definition is a coarse analogue of the notion of cohopfian groups from group theory.

Fact: a hyperbolic group 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is quasi-isometrically cohopfian iff 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is quasisymmetrically cohopfian, cf. MR1395067.

Problem 8.8 (Comments) [Ilia Kapovich] Take your favorite metric fractal. Is it quasisymmetrically cohopfian? What about the boundaries of hyperbolic groups?

Subproblem: What about the case of (round) Sierpinski carpets and Menger spaces which appear as boundaries of hyperbolic groups.

Background. Round Sierpinski carpets are the ones which are bounded by round circles. Such sets arise as the ideal boundaries of fundamental groups of compact hyperbolic manifolds with nonempty totally geodesic boundary. It is known that if 

dvipng error! exitcode was 2 (signal 0), transscript follows:
are such groups which are not commensurable then their ideal boundaries are not quasisymmetric to each other. There is a similar rigidity theorem (due to Marc Bourdon and Herve Pajot MR1789183) for a certain class of Menger curves, i.e. the ones which appear as visual boundaries of 2-dimensional Fuchsian buildings. Quasisymmetric cohopfian property is open in both cases.

Remark [Danny Calegari] As an example for the previous problem: the limit set 

dvipng error! exitcode was 2 (signal 0), transscript follows:
of a leaf of a taut foliation of a hyperbolic 3-manifold with 1-sided branching is a dendrite in 
dvipng error! exitcode was 2 (signal 0), transscript follows:
which is nowhere dense, has Assouad dimension 2, and for any point 
dvipng error! exitcode was 2 (signal 0), transscript follows:
in 
dvipng error! exitcode was 2 (signal 0), transscript follows:
and any neighborhood 
dvipng error! exitcode was 2 (signal 0), transscript follows:
of 
dvipng error! exitcode was 2 (signal 0), transscript follows:
in 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, 
dvipng error! exitcode was 2 (signal 0), transscript follows:
can be embedded by a conformal automorphism of 
dvipng error! exitcode was 2 (signal 0), transscript follows:
into 
dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Remark [Juha Heinonen] If 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is the standard "square" Menger space 
dvipng error! exitcode was 2 (signal 0), transscript follows:
then it is clearly not quasisymmetrically cohopfian.

Problem 8.9 (Comments) [Conjecture: Juha Heinonen] If 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is Loewner, then it is quasisymmetrically cohopfian. (Boundaries of Fuchsian buildings provide a good test case for this conjecture.)

Problem 8.10 (Comments) [Bruce Kleiner] If 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is a hyperbolic group and 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is connected with no local cut points, is there a natural measure class which is quasisymmetrically invariant? (That is, invariant under quasisymmetric homeomorphisms 
dvipng error! exitcode was 2 (signal 0), transscript follows:
.)

Motivation: rigidity theorems rely on absolute continuity of quasisymmetric maps as a foundational ingredient.

Remark If 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is Loewner, then the answer is "yes." But there are examples of Bourdon and Pajot MR1979183 whose boundary is not Loewner for each metric which is quasisymmetric to a Gromov-type metric.

In Bourdon---Pajot examples, Patterson-Sullivan measure works because there are relatively few quasisymmetric maps.

Problem 8.11 (Comments) [Kim Ruane] Can you do analysis on CAT(0) boundaries? With no natural metric, is there any structure beyond topology?

Remark [Bruce Kleiner] "Pushing in" the visual sphere gives pseudo-metrics on 

dvipng error! exitcode was 2 (signal 0), transscript follows:
, where 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is the CAT(0) space acted on by 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. Consider the radial projection 
dvipng error! exitcode was 2 (signal 0), transscript follows:
to spheres of radius 
dvipng error! exitcode was 2 (signal 0), transscript follows:
; then 
dvipng error! exitcode was 2 (signal 0), transscript follows:
are the pseudo-metrics. But then for a function 
dvipng error! exitcode was 2 (signal 0), transscript follows:
going quickly enough to zero, 

dvipng error! exitcode was 2 (signal 0), transscript follows:

is a metric on 

dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Remark [Damian Osajda] One can define a family 

dvipng error! exitcode was 2 (signal 0), transscript follows:
of metrics on 
dvipng error! exitcode was 2 (signal 0), transscript follows:
as follows. Pick 
dvipng error! exitcode was 2 (signal 0), transscript follows:
and choose a base-point 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. Let 
dvipng error! exitcode was 2 (signal 0), transscript follows:
be geodesic rays emanating from 
dvipng error! exitcode was 2 (signal 0), transscript follows:
and asymptotic to points 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. Let 
dvipng error! exitcode was 2 (signal 0), transscript follows:
be such that 

dvipng error! exitcode was 2 (signal 0), transscript follows:

If such 

dvipng error! exitcode was 2 (signal 0), transscript follows:
does not exists (i.e. 
dvipng error! exitcode was 2 (signal 0), transscript follows:
), then set 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. Finally, set 

dvipng error! exitcode was 2 (signal 0), transscript follows:

Problem 8.12 (Comments) [Bruce Kleiner] 

dvipng error! exitcode was 2 (signal 0), transscript follows:
acts on 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. Is this action "nice" with respect to the metrics in the previous remark?

Problem 8.13 (Comments) [Marc Bourdon] If 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is the boundary of a hyperbolic group and 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is connected, has no local cut points, and is not Loewner, is there a quasisymmetrically invariant nontrivial closed equivalence relation 
dvipng error! exitcode was 2 (signal 0), transscript follows:
on 
dvipng error! exitcode was 2 (signal 0), transscript follows:
so that 
dvipng error! exitcode was 2 (signal 0), transscript follows:
is Hausdorff and is a boundary of 
dvipng error! exitcode was 2 (signal 0), transscript follows:
relative to a collection of parabolic subgroups?

Remark In Bourdon--Pajot examples MR1979183, the answer to the above problem is positive.

Problem 8.14 (Comments) [Jeremy Tyson] Study relationships between different notions of conformal structure on 

dvipng error! exitcode was 2 (signal 0), transscript follows:
for hyperbolic 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. Here is an (incomplete) list of such notions: 

  1. dvipng error! exitcode was 2 (signal 0), transscript follows:
    -quasiconformal in the metric sense, i.e. 
    dvipng error! exitcode was 2 (signal 0), transscript follows:
    \label{bonk}.
  2. preserving modulus of curves joining two compacta \label{modulus}. 

  3. dvipng error! exitcode was 2 (signal 0), transscript follows:
    -quasisymmetric with 
    dvipng error! exitcode was 2 (signal 0), transscript follows:
    as close to linear as we like. (
    dvipng error! exitcode was 2 (signal 0), transscript follows:
    are functions of the point 
    dvipng error! exitcode was 2 (signal 0), transscript follows:
    where we test 
    dvipng error! exitcode was 2 (signal 0), transscript follows:
    for conformality)

  4. if Poincar\'{e} inequality holds for 

    dvipng error! exitcode was 2 (signal 0), transscript follows:
    , then, using Cheeger cotangent bundle 
    dvipng error! exitcode was 2 (signal 0), transscript follows:
    , can give a notion of measurable bounded conformal structure 
    dvipng error! exitcode was 2 (signal 0), transscript follows:
    such that 

    dvipng error! exitcode was 2 (signal 0), transscript follows:
    is a convergence group.

Remark Good notions of quasiconformality should have the convergence property, and metric notion does not, so its usefulness would be if ~(\ref{bonk})

dvipng error! exitcode was 2 (signal 0), transscript follows:
~(\ref{modulus}), since ~(\ref{bonk}) is checkable and ~(\ref{modulus}) is not.

Recall that if 

dvipng error! exitcode was 2 (signal 0), transscript follows:
is a homeomorphism for 
dvipng error! exitcode was 2 (signal 0), transscript follows:
, then 

dvipng error! exitcode was 2 (signal 0), transscript follows:

Problem 8.15 (Comments) [Juha Heinonen] Is the same true for Hilbert spaces? (All known proofs of above type use geometry, not analysis.)

It follows from work of Mario Bonk and Oded Schramm that there are quasi-isometric embeddings of 

dvipng error! exitcode was 2 (signal 0), transscript follows:
(quaternionic hyperbolic space) into 
dvipng error! exitcode was 2 (signal 0), transscript follows:
which are very far from isometric embeddings. (One can construct such examples with 
dvipng error! exitcode was 2 (signal 0), transscript follows:
for a constant 
dvipng error! exitcode was 2 (signal 0), transscript follows:
which is no less than 16.)

Problem 8.16 (Comments) [David Fisher] Can one do this with smaller 

dvipng error! exitcode was 2 (signal 0), transscript follows:
? Say 
dvipng error! exitcode was 2 (signal 0), transscript follows:
? (Same problem valid for complex hyperbolic space.) \label{fisher}

Subproblem (Misha Kapovich): Consider 

dvipng error! exitcode was 2 (signal 0), transscript follows:
sitting inside of 
dvipng error! exitcode was 2 (signal 0), transscript follows:
. Is 
dvipng error! exitcode was 2 (signal 0), transscript follows:
locally quasi-symmetrically rigid in 
dvipng error! exitcode was 2 (signal 0), transscript follows:
? More precisely, is it true that each quasisymmetric embedding 
dvipng error! exitcode was 2 (signal 0), transscript follows:
which is sufficiently close to the identity is induced by an isometry of 
dvipng error! exitcode was 2 (signal 0), transscript follows:
?

Remark This subproblem might be easier to settle than Problem~\ref{fisher}, since one can try to use infinitesimal tools like quasiconformal vector-fields.

David Fisher and Kevin Whyte have constructed some "exotic" quasi-isometric embeddings for higher-rank symmetric spaces that are "algebraic", in the sense that 

dvipng error! exitcode was 2 (signal 0), transscript follows:
.

Problem 8.17 (Comments) [David Fisher] Are all quasi-isometric embeddings between higher-rank symmetric spaces either isometries or algebraic in this way?

Problem 8.18 (Comments) [Misha Kapovich] Let 

dvipng error! exitcode was 2 (signal 0), transscript follows:
be a hyperbolic group. Is it true that 
dvipng error! exitcode was 2 (signal 0), transscript follows:
admit a uniformly quasiconformal discrete action on 
dvipng error! exitcode was 2 (signal 0), transscript follows:
(for some 
dvipng error! exitcode was 2 (signal 0), transscript follows:
)?

The answer is probably negative. It is reasonable to expect that every group satisfying Property (T) which admits such an action must be finite. However the usual proofs that infinite discrete subgroups of 

dvipng error! exitcode was 2 (signal 0), transscript follows:
never satisfy Property (T) do not work in the quasiconformal category.

Next Section

Boundaries/Section8 (last edited 2010-09-15 23:15:25 by RickScott)