Research Statement. Jeffrey Lindquist

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1 Research Statement Jeffrey Lindquist 1 Introduction My areas of interest and research are geometric function theory, geometric group theory, and analysis on metric spaces. From the pioneering complex analysis work of Ahlfors [Ah] to the modulus developments by Beurling and Fuglede to the rigidity theorem of Mostow [Mo], smooth geometry has been studied extensively. More recently, there has been a desire to generalize results of this nature to a nonsmooth setting. Books such as [He] and [HKST] further this goal: both work in metric measure spaces with few assumed regularity conditions. Tools to deal with such spaces have been developed, for instance, in [HK]. One area that drives this research is geometric group theory. Gromov s work on hyperbolic spaces has provided a geometric link to groups that has been extremely valuable in their study: understanding the geometry of hyperbolic spaces and their boundaries leads to stronger understanding of hyperbolic groups. This connection allows hyperbolic group questions to be phrased in terms of nonsmooth analysis: Cannon s conjecture on the boundaries of hyperbolic groups, for instance, can be rephrased in terms of quasisymmetric equivalence of certain topological 2-spheres with S 2. This viewpoint provides substantial progress towards Cannon s conjecture in [BK]. My research fits into this framework. My thesis work centers around the use of hyperbolic fillings to define extensions of the modulus of path families to suitably nice metric spaces. Hyperbolic fillings are graphs which encode the combinatorial geometry of a metric space on many scales. These fillings are analogous to Cayley graphs in the geometric group setting: given a finitely generated group one can form its Cayley graph which, if it is a Gromov hyperbolic metric space, has an associated boundary at infinity metric space that is compact and Ahlfors regular. Turning this around, if we instead start with a compact, Ahlfors regular metric space Z, we can fill in a hyperbolic filling that is a Gromov hyperbolic metric space with Z as the boundary at infinity. The construction used in my thesis is very similar to the constructions in [BP] and [BuS]. A useful picture to have in mind is that of a Whitney cube decomposition of the hyperbolic plane H 2 where cubes correspond to vertices. In this setting, two cubes are connected by an edge if their boundaries intersect. The modulus extensions I considered have quasisymmetric invariance properties. A homeomorphism ϕ : Z W between two metric spaces is called a quasisymmetry if there is a homeomorphism η : [0, ) [0, ) such that for all a, b, c Z, a b t a c = ϕ(a) ϕ(b) η(t) ϕ(a) ϕ(c). The intuition here is that quasisymmetries distort relative shapes by a controlled amount (much like quasiconformal maps in the classical setting). One can show that if ϕ is a quasisymmetry then ϕ 1 is as well. Thus, metric spaces are partitioned into equivalence classes with two spaces being equivalent if there is a quasisymmetric map between them. Classifying these equivalence classes still poses challenges and, as noted above, is desirable for proving results such as Cannon s conjecture. 1

2 The groundwork for the modulus generalizations is developed in [L1]. In this paper I define the modulus extensions, prove comparability results with modulus and quasisymmetric invariance, and begin to investigate some critical exponents that can be defined. As an application, one can prove a result in [Ty], namely that a quasisymmetry between two Ahlfors Q-regular metric spaces preserves the Q-modulus of path families up to a multiplicative constant depending only on the quasisymmetry. Tyson makes use of modulus generalizations related to [P1] and [P2] to prove this result, and Pansu s work is foundational in what I have done as well. In [L2], the question of how rigid hyperbolic fillings are is addressed. While it was known that two hyperbolic fillings of the same metric space must be quasi-isometric, it turns out that the vertex sets of these fillings are actually bilipschitz equivalent. This result follows from a more general quasi-isometric promotion criterion from [Wh]. Corollary (L.). Let (Z, d Z ) and (W, d W ) be quasisymmetrically equivalent, compact, Ahlfors regular metric spaces. Let X = (V X, E X ) and Y = (V Y, E Y ) be hyperbolic fillings of Z and W. Then, V X and V Y are bilipschitz equivalent. In [L3] I investigate critical exponents related to one of the modulus generalizations, weak capacity, in more depth. These critical exponents lead to a version of a result in [BK] on necessary and sufficient conditions for a topological 2-sphere to be quasisymmetrically equivalent to S 2. The generalized version of the result in [BK] is used to prove the following corollary. Corollary (L.). Let (Z, d, µ) be an Ahlfors Q-regular metric measure space homeomorphic to S 2 that is linearly locally connected. Suppose there is a metric space Y quasi-isometric to a hyperbolic filling X of Z and a group G Iso(Y ) acting cocompactly on Y. Then, there exist finitely many pairs of sets (A i, B i ) with the property that if wcap 2 (A i, B i ) < for all i, then Z is quasisymmetrically equivalent to S 2. A group action of G on Y is cocompact if there is a compact set K such that for every y Y, there is an element g G with gy K. Here Iso(Y ) refers to the group of isometries of Y. My current work involves the study of quasiregular maps which are the noninjective counterpart of quasiconformal maps. These maps serve as suitable generalizations of conformal maps, particularly in higher dimensions where the class of conformal maps is very rigid. I am interested in extension properties of these maps as well as the question of if they can be viewed from the hyperbolic filling perspective that was successful in my thesis. I am also investigating the connection between weak capacity and various notions of Sobolev capacity as inspired by [BoS]. This involves modifying techniques used in my thesis to apply to more general sets than just open set pairs and continuua. 2 Details I will now provide a more detailed overview of my research. In what follows we assume all metric measure spaces are compact, connected, and Ahlfors Q-regular. The regularity condition says that the measure of any ball of radius r 1 is, up to a fixed multiplicative constant, r Q. To create a hyperbolic filling of a metric space Z, one chooses a scale s > 1 and, for each n N 0, a set of s n -separated points in Z that is maximal under inclusion. One then uses these points as the centers of balls of radius 2s n. These balls form the vertices on level n. Connecting intersecting vertices on neighboring levels and imbuing the graph with the natural path metric completes the 2

3 construction. One can show that if ϕ : Z W is a quasisymmetry and X and Y are hyperbolic fillings of Z and W respectively, then ϕ induces a map φ : X Y which is a quasi-isometry. This means there is a constant C > 0 such that for all x, x X, we have C 1 x x C φ(x) φ(y) C x x + C and that every point in y is within C of some point in φ(x). This fact forms the foundation for the quasisymmetric invariance of the modulus generalizations in my research. There are two approaches to extending modulus in my work, although one of them has received more focus. Both of these are modeled on the definition of modulus. Instead of paths in the original space, however, paths in the hyperbolic filling are used. Traditional modulus is defined as follows: given a family Γ of paths, a Borel function ρ : Z [0, ] is admissible for Γ if for all locally rectifiable γ Γ we have ρ 1 (where all path integrals are with respect to the arclength γ parameterization). The p-modulus of the path family Γ is then inf Z ρp where the infimum is taken over all admissible ρ. The main generalization, called weak capacity and denoted wcap p, is a quantity defined for two sets A and B. In practice, it has been best to use sets A and B that are either both open with dist(a, B) > 0 (called positive separation) or disjoint continua. Here by a continuum we mean a compact, connected set with more than two points. In nice settings (e.g. S n ), the quantity wcap p (A, B) is comparable to the modulus of the right dimension (n in the case of S n ) of the family of paths connecting A to B in Z. Let A and B be two open sets with positive separation or disjoint continua. For admissibility, we consider functions defined on edges of a given hyperbolic filling X and infinite paths with limits in A and B. We call such a function τ admissible if τ(e) 1 for all of these infinite paths. We then define the (weak) p-capacity of A and B as wcap p (A, B) = inf τ p p, where the infimum is taken over all admissible τ and p, denotes the weak l p -norm. The weak norm of a function f : X C defined on a countable space X is the infimum of C such that #{x : f(x) > λ} Cp λ p for all λ > 0. Strictly speaking this is not a norm, but it is comparable to one provided that p > 1. The use of the weak norm here and not the usual l p norm is motivated by [BoS]. We note we can use vertex functions instead of edge functions. The second generalization relies on approximating paths on the boundary with paths in the hyperbolic filling. As this generalization has received less attention, we will not go through the details here. We only note that it is this quantity that allows us to reprove the result in [Ty]. In [L1] it is shown that these generalizations have a quasisymmetric invariance property. The more difficult part comes in showing that the generalizations actually are modulus extensions. That is, that these quantities are comparable to the standard p-modulus under the appropriate conditions. One thing to note is that wcap p does not rely on rectifiable paths whereas modulus does. For this reason we should expect that in general there is a constant C such that for all open A, B with dist(a, B) > 0 or disjoint continua we have mod p (A, B) C wcap p (A, B) (here mod p (A, B) denotes the p-modulus of the path family connecting A to B). For the other direction one needs a way to guarantee the existence of many rectifiable paths. For this, the Loewner condition from [HK] is used. 3

4 Theorem (L.). Let Q > 1 and let (Z, d, µ) be a compact, connected Ahlfors Q-regular metric space which is also a Q-Loewner space. Then there exist constants c, C > 0 depending only on Q and the hyperbolic filling parameters with the following property: whenever A, B Z are either open sets with positive separation or disjoint continua, c mod Q (A, B) wcap Q (A, B) C mod Q (A, B). Note Q is used here in place of p as it is important to match the Ahlfors regularity dimension. The proofs of the comparability results follow the same scheme: given an admissible function for the presumed larger quantity, construct an admissible function for the smaller quantity. Showing the mass of the constructed function is controlled by the mass of the original function then proves the result. While the proof strategy is straightforward, some of the details rely on some difficult results. In particular, the fact that the Poincaré inequality is an open condition as proven in [KZ] is used for the Loewner direction and a maximal function estimate due to [BoS] is used for the mass bounds of the subsequential u n limit. Using weak capacity, we can define critical exponents which are quasisymmetric invariants of the metric space in question. One way to do this is to look at the infimum of p such that wcap p (A, B) < for all open sets A and B with positive separation. Another way is to require control of wcap p by relative distance: we look at the infimum of p such that the function ϕ p (t) = sup{wcap p (A, B) : (A, B) t} is finite for all open A and B with positive separation. Here (A, B) = dist(a, B) min(diam(a), diam(b)) is the relative distance between A and B. We can prove a uniformization result as in [BK] for weak capacity. Theorem (L.). Let (Z, d, µ) be an Ahlfors Q-regular metric measure space homeomorphic to S 2 that is linearly locally connected. Suppose there exists t 0 > 0 such that ϕ 2 (t 0 ) <. Then, Z is quasisymmetrically equivalent to S 2. Conversely, if Z is quasisymmetrically equivalent to S 2, then ϕ 2 (t) is finite for all t and ϕ 2 (t) 0 as t. Linear local connectivity is a technical condition that is required for quasisymmetric equivalence with S 2. It turns out that the two critical exponents defined above agree in the hyperbolic group setting. In the following, these exponents are denoted Q w and Q w and Iso(Y ) refers to the group of isometries of Y. Theorem (L.). Let (Z, d, µ) be a compact, connected, Ahlfors regular metric measure space. Suppose there is a metric space Y quasi-isometric to a hyperbolic filling X of Z and a group G Iso(Y ) acting cocompactly on Y. Then, Q w (Z) = Q w(z). If the infimum is attained in the definition of Q w, then the infimum is attained in the definition of Q w. Combining these results, we get the following characterization in the introduction. These results and more study of Q w and Q w can be found in [L3]. 4

5 3 Future Work With regards to weak capacity, there are a number of options for future development. Right now I am pursuing studying weak capacity in the context of a capacity (as opposed to as a modulus generalization) with Eero Saksman and investigating properties of sets more general than open sets with positive separation and continuua. Question. What properties of capacities does weak capacity have? Does weak capacity reduce to Sobolev capacity in nice settings? This is related to the work in [BoS] and [BP]. In [BoS], a Sobolev space using the weak norm on a hyperbolic filling is developed. In [BP], the l p norm on a hyperbolic filling is used and connected to Besov spaces on the boundary. In [BouK] the Combinatorial Loewner Property (CLP) is discussed. From [BouK, Section 3.2] a compact, arcwise connected, doubling metric space satisfies the Combinatorial p-loewner Property if there exists two positive, increasing functions φ, ψ : (0, ) (0, ) with lim t 0 ψ(t) = 0 such that, for all disjoint non-degenerate continua A, B and k large enough, one has φ( (A, B) 1 ) mod p (A, B, G k ) ψ( (A, B) 1 ) where G k is a κ-approximation of the metric space as defined in the paper. We can define a weak capacity Loewner Property analogously: instead of using κ-approximations and mod p (A, B, G k ), we use wcap p (A, B). It is known that a Loewner space always satisfies the CLP. Question. How does the weak capacity Loewner Property relate to the Loewner Property and the Combinatorial Loewner property? The quasisymmetric invariance of wcap p illustrates that it may be useful in many other contexts in which quasisymmetric maps are present. The quasisymmetric recognition problem asks when two given spaces are quasisymmetrically equivalent. By comparing wcap p on the two spaces, one may be able to show two spaces are not equivalent. There is particular interest in this when one of the spaces is S 2. Another situation to look for applications of wcap p or a suitable modification is when the set up involves two spaces, one acting as a boundary at infinity for the other. For example, one may speculate connections with expanding Thurston maps (for instance, [BM, Theorem 1.9]) as these have associated tilings which behave like hyperbolic fillings. With regards to quasiregular mappings, I am currently working with Pekka Pankka to investigate the effect a quasiregular map has on a hyperbolic filling. This insight looks promising for extension results and, as the quasisymmetry/quasi-isometry connection has proven so fruitful, will likely given greater insight into quasiregular maps. This study also gives rise to classes of maps that may lie between quasiregular maps and quasiconformal maps or quasisymmetric maps. Question. Can one define a class of functions on hyperbolic fillings that extend quasiregular maps in a similar way that quasi-isometric maps extend quasisymmetric maps? What are the properties of these maps? References [Ah] L. Ahlfors, Conformal invariants: topics in geometric function theory, American Mathematical Soc (2010). 5

6 [BK] M. Bonk and B. Kleiner, Quasisymmetric parameterizations of two-dimensional metric spheres, Invent. Math. 150 (2002), [BM] M. Bonk and D. Meyer, Expanding Thurston maps, arxiv preprint [BouK] M. Bourdon and B. Kleiner, Combinatorial modulus, the Combinatorial Loewner Property, and Coxeter groups, Groups Geom. Dyn. 7 (2013), no. 1, [BoS] M. Bonk and E. Saksman, Sobolev spaces and hyperbolic fillings, to appear in: J. Reine Angew. Math. (arxiv preprint ). [BP] M. Bourdon and H. Pajot, Cohomologie l p et espaces de Besov, J. Reine Angew. Math. 558 (2003), [BuS] S. Buyalo and V. Schroeder, Elements of asymptotic geometry, Europ. Math. Soc., Zürich, [He] J. Heinonen, Lectures on analysis on metric spaces, Springer, New York, [HK] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), [HKST] J. Heinonen, P. Koskela, N. Shanmugalingam, and J. Tyson, Sobolev Spaces on Metric Measure Spaces, Cambridge University Press, Cambridge, [KZ] S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. (2) 167 (2008), [L1] [L2] [L3] J. Lindquist, Weak capacity and modulus comparability in Ahlfors regular metric spaces, Analysis and Geometry in Metric Spaces, 4, (2016) no. 1, J. Lindquist, Bilipschitz equivalence of trees and hyperbolic fillings (in preparation, current version at J. Lindquist, Weak capacity and critical exponents (in preparation, current version at LINK!!) [Mo] G. Mostow, Quasi-conformal mappings in n-space and the rigidity of the hyperbolic space forms, Publ. Math. IHES 34 (1968), [P1] [P2] [Ty] P. Pansu, Métriques de Carnot-Carathodory et quasiisométries des espaces symétriques de rang un. Ann. of Math. (2) 129 (1989), P. Pansu, Dimension conforme et sphère à l infini des variétés à courbure négative. Ann. Acad. Sci. Fenn., Ser. A I, 14, (1989), J. Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 23 (1998), [Wh] K. Whyte, Amenability, bilipschitz equivalence, and the von Neumann conjecture, Duke Math. J. 99, (1999) no. 1,