# Hyperbolic geometry and quasiconformal mappings (05frg502)

Arriving in Banff, Alberta Saturday, August 6 and departing Saturday August 13, 2005

## Organizers

Petra Bonfert-Taylor (Wesleyan University)

Martin Bridgeman (Boston College)

Dick Canary (University of Michigan)

Gaven Martin (Massey University)

Richard Schwartz (University of Maryland)

Edward Taylor (Wesleyan University)

## Objectives

As stated above, we propose a research stay at BIRS with the following objective: We wish to assemble a broad and diverse working group in hyperbolic geometry and conformal analysis in order to review and extend the current state of knowledge in applications of quasiconformal mappings to hyperbolic geometry as given in the three topics listed above. We give certain specific examples of questions which we hope to investigate during our proposed stay.

I.) An exciting aspect of this melding of geometry, topology and analysis has been isolated in the form of the quasiconformal homogeneity of a hyperbolic manifold. We recall that a hyperbolic n-manifold Mn is quasiconformally homogeneous if, for any two points x and y in Mn, there exists a quasiconformal automorphism of Mn that maps x to y. We say Mn is uniformly quasiconformally homogeneous if there exists a constant K > 1 so that there for any two points x; y 2 Mn there exists a K-quasiconformal automorphism of Mn that takes x to y. This notion was originally formulated by F. Gehring and B. Palka in the context of domains in Rn, and was recently extended to hyperbolic manifolds via work joint work of P. Bonfert-Taylor, R. Canary, G. Martin and E. Taylor. It is known that the quasiconformal homogeneity of a hyperbolic manifold is revealing of its geometry and topology: In dimensions n ¸ 3, a hyperbolic manifold N is uniformly quasiconformally homogeneous if and only if it is the regular cover of a closed hyperbolic orbifold.

a.) As a consequence of the Sullivan Rigidity Theorem we have shown, in dimension n ¸ 3, that there is a constant K(n) > 1 so that the following is true: If Mn 6= Hn is a hyperbolic n-manifold that is K-quasiconformally homogeneous, then K ¸ K(n). In other words, if a hyperbolic n-manifold is K-quasiconformally homogeneous for some K < K(n), then the manifold is all of Hn. The Sullivan Rigidity Theorem is not true in dimension two. Thus the first question is whether such a K(2) exists. That is, it is unknown whether in dimension n = 2 there exists such a constant K(2) > 1.

b.) We recall that in dimensions n ¸ 3 there is a coarse classification of uniformly quasiconformally homogenous hyperbolic manifolds. However, in dimension n = 2, any possible classification conjectures is known to be much more complex via the construction of examples. For instance, even in the setting of planar domains no complete classification is known. F. Gehring and B. Palka have shown that if D is a multiply connected uniformly quasiconformally homogenous domain in ^C with more than two boundary points, then every neighborhood of each boundary point of D contains infinitely many components of the complement of D. On the other hand, it is not hard to see that the complement of the Cantor ternary set in ^C and any quasiconformal image of this domain are uniformly quasiconformally homogeneous. In particular, this implies that any uniform domain whose boundary is a uniformly perfect, compact, totally disconnected set is uniformly quasiconformally homogenous.

An important goal for us during our proposed research week at Ban® is to explore possible classification schemes for both planar domains, and more generally, hyperbolic 2-manifolds that are uniformly quasiconformally homogeneous. c.) Finally, we restrict ourselves to closed hyperbolic surfaces. If one fixes the genus g of the surfaces under consideration, then using distance distortion estimates for quasiconformal mappings, as well as the Deligne-Mumford compactness theorem on Moduli space, one can observe that within a fixed moduli space Mg there does exist a constant Kg with the following property: If S is a closed hyperbolic surface of genus g ¸ 2 then S is not K-quasiconformally homogeneous for any K < Kg. It would be nice to show that a surface that attains the number Kg for its quasiconformal homogeneity constant in fact is the ?most symmetric within? its genus. More precisely, we conjecture: a structure in Mg that is Kg quasiconformally homogeneous possesses a large conformal automorphism group. (It is a result of R. Accola that there exists for each genus g there exist complex structures that admit at least 8(g + 1) conformal automorphisms; it is classical that an automorphism group has size less than or equal to 84(g ¡ 1).)

These questions involve understanding a broad set of phenomena within geometric analysis, Teichm¨uller spaces, complex geometry, and the geometric action of the mapping class group on Teichm¨uller space. We believe that the first question is ripe for resolution, and the assembly of this group may well be able to push this through. It strikes us that the set of questions grouped in (2) and (3) are more subtle. Thus our second objective is to focus on the delineation of issues surrounding resolving these questions, and to probe various preliminary approaches currently under investigation by members of the working group.

II.) Let ¡ be a convex co-compact torsion-free Kleinian group so that the quotient of Hn=¡ is of infinite volume. The Patterson-Sullivan geodesic current is the maximally ergodic ¡-invariant measure on the space of (unoriented) geodesics, each of whose endpoints lie in the limit set of ¡. Following work of F. Bonahon, M. Bridgeman and E. Taylor have in a series of papers related this measure to the quasiconformal deformation theory of such groups, in particular we have shown that the analytic distortion of the Hausdor® dimension of the limit sets of two quasiconformally conjugate convex co-compact Kleinian groups is bounded by the induced quasiconformal distortion of the Patterson-Sullivan measures. This knowledge represents a new tool with which to probe the geometric deformation of a convex co-compact Kleinian group under quasiconformal deformation. Using the ideas surrounding this result there are two particular questions in which we wish to focus:

a.) Following joint work of R. Canary, Y. Minksy and E. Taylor it is conjectured that if M is a compact, hyperbolizable and acylindrical 3-manifold, then the hyperbolic structure exhibiting minimum Hausdor® of its limit set is the (unique) hyperbolic structure in the quasiconformal deformation space whose convex core boundary is totally geodesic. We believe we can make partial progress on this conjecture, using our techniques as well as recent work of U. Hammenstadt, by showing that the Hausdor® dimension function has a local minimum at the acylindrical hyperbolic structure whose convex core boundary is totally geodesic. We note that a directly analogous question concerning the convex core volume function has been answered in the positive by P. Storm.

b.) An initial motivation for the undertaking of our study was to discover whether a Weil-Petersson type metric could be developed on general quasiconformal deformation spaces by using analogues of the techniques of F. Bonahon?s construction of the Weil-Petersson metric on Teichm¨uller space. Recent progress has been made on realizing this goal, though (apparently significant) analytic obstacles remain. We(M. Bridgeman and E. Taylor) wish to give a detailed report to the assembled experts both on our progress and on the remaining obstacles towards the completion of this project.

III.) We wish to explore the distortion of the Hausdor® dimension of limit sets of Kleinian groups and their exponents of convergence under quasisymmetric conjugation in dimension 1. In higher dimensions many of the questions below have been answered in joint work of P. Bonfert-Taylor, M. Bridgeman, and E. Taylor. Provocatively, our knowledge of this phenomena in dimension n = 1 has many gaps. The concept of quasiconformality in dimension n = 1 is too restrictive, and so this leads us to consideration of quasisymmetric maps. Informally, a quasisymmetric map in dimension one has the property that it quasi-preserves ratios. This concept extends to both higher dimensions and more general metric spaces. We note that while in dimensions n ¸ 2, a homeomorphism Á : Rn ! Rn is quasisymmetric if and only if it is quasiconformal, in dimension n = 1 the class of quasisymmetric homeomorphisms of R is a strictly larger class.

Quasisymmetric maps of the real line can be measure-theoretically very singular. Though quasiconformal mappings in Rn; n ¸ 2 preserve null-sets and sets of Hausdor® dimension n, neither of these facts are true for quasisymmetric maps of the real line. In particular, a subset of the real line of dimension strictly less than one can be mapped under a quasisymmetric mapping of R onto a set of dimension one. This cannot happen however for a dynamically interesting class of sets - the limit sets of purely conical discrete quasisymmetric groups. In fact, we have shown that limit sets of such groups are porous, and since porosity is a quasisymmetric invariant, we conclude that the dimension being smaller than one is a quasisymmetric invariant for the limit sets of two quasisymmetrically conjugate discrete quasisymmetric convergence groups.

Let G be a discrete quasisymmetric group as above, and let Á be a quasisymmetric mapping. We want to find explicit bounds on dim Á(¤(G)) in terms of dim ¤(G) and the quasisymmetry constant. A conjectural program exists to provide such bounds, and we outline below two of the most interesting questions associated to this program.

a.) We ask: Under what assumptions is the porosity constant of the limit set uniquely determined by its dimension? We note that this question is inherently of a 1-dimensional nature. If one naturally extends a Fuchsian group G as above to a group acting on R2, then the limit set of G (being contained in R ½ R2) is 1-porous in R2, and hence the porosity constant carries little information about the dimension of the limit set.

b.) One important assumption in the above question should be for the local porosity constant to be fixed on ¤(G). In fact, we ask: Let A ½ R be a set for which there exists ®0 2 (0; 1) so that for all a 2 A,

supf® > 0 jA is ® ¡ porous at ag = ®0:

Then is dimA = log(2)= log((2 ¡ ®0)=(1 ¡ ®0))?

We expect that under the assumption that G?s limit set be purely conical the local porosity constant of ¤(G) will be constant on ¤(G).

A positive resolution of (a) and (b) would link the Hausdor® dimension of limit sets of purely conical Fuchsian groups to their porosity constant. Thus the study of the distortion of Hausdor® dimension for these sets reduces to the study of distortion of porosity which is an easier question since porosity is a general quasisymmetric invariant for all subsets of R.

I.) An exciting aspect of this melding of geometry, topology and analysis has been isolated in the form of the quasiconformal homogeneity of a hyperbolic manifold. We recall that a hyperbolic n-manifold Mn is quasiconformally homogeneous if, for any two points x and y in Mn, there exists a quasiconformal automorphism of Mn that maps x to y. We say Mn is uniformly quasiconformally homogeneous if there exists a constant K > 1 so that there for any two points x; y 2 Mn there exists a K-quasiconformal automorphism of Mn that takes x to y. This notion was originally formulated by F. Gehring and B. Palka in the context of domains in Rn, and was recently extended to hyperbolic manifolds via work joint work of P. Bonfert-Taylor, R. Canary, G. Martin and E. Taylor. It is known that the quasiconformal homogeneity of a hyperbolic manifold is revealing of its geometry and topology: In dimensions n ¸ 3, a hyperbolic manifold N is uniformly quasiconformally homogeneous if and only if it is the regular cover of a closed hyperbolic orbifold.

a.) As a consequence of the Sullivan Rigidity Theorem we have shown, in dimension n ¸ 3, that there is a constant K(n) > 1 so that the following is true: If Mn 6= Hn is a hyperbolic n-manifold that is K-quasiconformally homogeneous, then K ¸ K(n). In other words, if a hyperbolic n-manifold is K-quasiconformally homogeneous for some K < K(n), then the manifold is all of Hn. The Sullivan Rigidity Theorem is not true in dimension two. Thus the first question is whether such a K(2) exists. That is, it is unknown whether in dimension n = 2 there exists such a constant K(2) > 1.

b.) We recall that in dimensions n ¸ 3 there is a coarse classification of uniformly quasiconformally homogenous hyperbolic manifolds. However, in dimension n = 2, any possible classification conjectures is known to be much more complex via the construction of examples. For instance, even in the setting of planar domains no complete classification is known. F. Gehring and B. Palka have shown that if D is a multiply connected uniformly quasiconformally homogenous domain in ^C with more than two boundary points, then every neighborhood of each boundary point of D contains infinitely many components of the complement of D. On the other hand, it is not hard to see that the complement of the Cantor ternary set in ^C and any quasiconformal image of this domain are uniformly quasiconformally homogeneous. In particular, this implies that any uniform domain whose boundary is a uniformly perfect, compact, totally disconnected set is uniformly quasiconformally homogenous.

An important goal for us during our proposed research week at Ban® is to explore possible classification schemes for both planar domains, and more generally, hyperbolic 2-manifolds that are uniformly quasiconformally homogeneous. c.) Finally, we restrict ourselves to closed hyperbolic surfaces. If one fixes the genus g of the surfaces under consideration, then using distance distortion estimates for quasiconformal mappings, as well as the Deligne-Mumford compactness theorem on Moduli space, one can observe that within a fixed moduli space Mg there does exist a constant Kg with the following property: If S is a closed hyperbolic surface of genus g ¸ 2 then S is not K-quasiconformally homogeneous for any K < Kg. It would be nice to show that a surface that attains the number Kg for its quasiconformal homogeneity constant in fact is the ?most symmetric within? its genus. More precisely, we conjecture: a structure in Mg that is Kg quasiconformally homogeneous possesses a large conformal automorphism group. (It is a result of R. Accola that there exists for each genus g there exist complex structures that admit at least 8(g + 1) conformal automorphisms; it is classical that an automorphism group has size less than or equal to 84(g ¡ 1).)

These questions involve understanding a broad set of phenomena within geometric analysis, Teichm¨uller spaces, complex geometry, and the geometric action of the mapping class group on Teichm¨uller space. We believe that the first question is ripe for resolution, and the assembly of this group may well be able to push this through. It strikes us that the set of questions grouped in (2) and (3) are more subtle. Thus our second objective is to focus on the delineation of issues surrounding resolving these questions, and to probe various preliminary approaches currently under investigation by members of the working group.

II.) Let ¡ be a convex co-compact torsion-free Kleinian group so that the quotient of Hn=¡ is of infinite volume. The Patterson-Sullivan geodesic current is the maximally ergodic ¡-invariant measure on the space of (unoriented) geodesics, each of whose endpoints lie in the limit set of ¡. Following work of F. Bonahon, M. Bridgeman and E. Taylor have in a series of papers related this measure to the quasiconformal deformation theory of such groups, in particular we have shown that the analytic distortion of the Hausdor® dimension of the limit sets of two quasiconformally conjugate convex co-compact Kleinian groups is bounded by the induced quasiconformal distortion of the Patterson-Sullivan measures. This knowledge represents a new tool with which to probe the geometric deformation of a convex co-compact Kleinian group under quasiconformal deformation. Using the ideas surrounding this result there are two particular questions in which we wish to focus:

a.) Following joint work of R. Canary, Y. Minksy and E. Taylor it is conjectured that if M is a compact, hyperbolizable and acylindrical 3-manifold, then the hyperbolic structure exhibiting minimum Hausdor® of its limit set is the (unique) hyperbolic structure in the quasiconformal deformation space whose convex core boundary is totally geodesic. We believe we can make partial progress on this conjecture, using our techniques as well as recent work of U. Hammenstadt, by showing that the Hausdor® dimension function has a local minimum at the acylindrical hyperbolic structure whose convex core boundary is totally geodesic. We note that a directly analogous question concerning the convex core volume function has been answered in the positive by P. Storm.

b.) An initial motivation for the undertaking of our study was to discover whether a Weil-Petersson type metric could be developed on general quasiconformal deformation spaces by using analogues of the techniques of F. Bonahon?s construction of the Weil-Petersson metric on Teichm¨uller space. Recent progress has been made on realizing this goal, though (apparently significant) analytic obstacles remain. We(M. Bridgeman and E. Taylor) wish to give a detailed report to the assembled experts both on our progress and on the remaining obstacles towards the completion of this project.

III.) We wish to explore the distortion of the Hausdor® dimension of limit sets of Kleinian groups and their exponents of convergence under quasisymmetric conjugation in dimension 1. In higher dimensions many of the questions below have been answered in joint work of P. Bonfert-Taylor, M. Bridgeman, and E. Taylor. Provocatively, our knowledge of this phenomena in dimension n = 1 has many gaps. The concept of quasiconformality in dimension n = 1 is too restrictive, and so this leads us to consideration of quasisymmetric maps. Informally, a quasisymmetric map in dimension one has the property that it quasi-preserves ratios. This concept extends to both higher dimensions and more general metric spaces. We note that while in dimensions n ¸ 2, a homeomorphism Á : Rn ! Rn is quasisymmetric if and only if it is quasiconformal, in dimension n = 1 the class of quasisymmetric homeomorphisms of R is a strictly larger class.

Quasisymmetric maps of the real line can be measure-theoretically very singular. Though quasiconformal mappings in Rn; n ¸ 2 preserve null-sets and sets of Hausdor® dimension n, neither of these facts are true for quasisymmetric maps of the real line. In particular, a subset of the real line of dimension strictly less than one can be mapped under a quasisymmetric mapping of R onto a set of dimension one. This cannot happen however for a dynamically interesting class of sets - the limit sets of purely conical discrete quasisymmetric groups. In fact, we have shown that limit sets of such groups are porous, and since porosity is a quasisymmetric invariant, we conclude that the dimension being smaller than one is a quasisymmetric invariant for the limit sets of two quasisymmetrically conjugate discrete quasisymmetric convergence groups.

Let G be a discrete quasisymmetric group as above, and let Á be a quasisymmetric mapping. We want to find explicit bounds on dim Á(¤(G)) in terms of dim ¤(G) and the quasisymmetry constant. A conjectural program exists to provide such bounds, and we outline below two of the most interesting questions associated to this program.

a.) We ask: Under what assumptions is the porosity constant of the limit set uniquely determined by its dimension? We note that this question is inherently of a 1-dimensional nature. If one naturally extends a Fuchsian group G as above to a group acting on R2, then the limit set of G (being contained in R ½ R2) is 1-porous in R2, and hence the porosity constant carries little information about the dimension of the limit set.

b.) One important assumption in the above question should be for the local porosity constant to be fixed on ¤(G). In fact, we ask: Let A ½ R be a set for which there exists ®0 2 (0; 1) so that for all a 2 A,

supf® > 0 jA is ® ¡ porous at ag = ®0:

Then is dimA = log(2)= log((2 ¡ ®0)=(1 ¡ ®0))?

We expect that under the assumption that G?s limit set be purely conical the local porosity constant of ¤(G) will be constant on ¤(G).

A positive resolution of (a) and (b) would link the Hausdor® dimension of limit sets of purely conical Fuchsian groups to their porosity constant. Thus the study of the distortion of Hausdor® dimension for these sets reduces to the study of distortion of porosity which is an easier question since porosity is a general quasisymmetric invariant for all subsets of R.