F. W. Gehring

The

Jf(x) = lim sup m(f(B(x, r)))/m(B(x, r)), r->0 where B(x, r) denotes the open ^-dimensional ball of radius r about x and m denotes Lebesgue measure in R. We call Lf(x) and Jf(x\ respectively, the maximum stretching and generalized Jacobian for the homeomorphism ƒ at the point x. These functions are nonnegative and measurable in D, and Lebesgues theorem implies that Jf is locally LMntegrable there. Suppose in addition that ƒ is X-quasiconformal in D. Then Lf ^ KJf a.e. in D, and thus Lf is locally L -integrable in D. Bojarski has shown in [1] that a little more is true in the case where n = 2, namely that Lf is locally L-integrable in D for p e [2, 2 + c), where c is a positive constant which depends only on K. His proof consists of applying the CalderonZygmund inequality [2] to the Hubert transform which relates the complex derivatives of a normalized plane quasiconformal mapping. Unfortunately this elegant two-dimensional argument does not suggest what the situation is when n > 2. The purpose of this note is to announce the following n-dimensional version of Bojarskis theorem. THEOREM. Suppose that D is a domain in R and that f\D^Risa K-quasiconformal mapping. Then Lf is locally L -integrable in D for p e [1, n + c\ where c is a positive constant which depends only on K and n.

L^p

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-integrability of the partial derivatives of a quasiconformal mapping

We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of P SL(2, C). We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between elliptic axes. We establish the discreteness and arithmeticity of a number of these extremal groups, the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple.

The coefficients of quasiconformality of domains in space

Suppose that D is a domain in the complex plane. Then D is said to have the Hardy-Littlewood property if there exists a constant c such that for 0 < k  1f is in the class Lipλ(D) with whenever f is analytic in D with in D Next D is said to be a K-quasidisk if it is the image of the unit disk under a K-quasiconformal mapping of the extended complex plane. If D is a quasidisk, then D has the Hardy—Littlewood property. Moreover if D is simply connected with ∞∊∂D, then D is a quasidisk if and only if D and its exterior D ∗ are domains which have the Hardy—Littlewood property.

Arithmeticity, discreteness and volume

(1989). Stability and extremality in j⊘rgensens inequality. Complex Variables, Theory and Application: An International Journal: Vol. 12, No. 1-4, pp. 277-282.

Quasidisks and the hardy—littlewood property

Extending earlier work, we establish the finiteness of the number of two-generator arithmetic Kleinian groups with one generator parabolic and the other either parabolic or elliptic. We also identify all the arithmetic Kleinian groups generated by two parabolic elements. Surprisingly, there are exactly 4 of these, up to conjugacy, and they are all torsion free.

Stability and extremality in j⊘rgensen's inequality

We find all real points of the analytic space of two generator Mobius groups with one generator elliptic of order two. Geometrically this is a certain slice through the space of two generator discrete groups, analogous to the Riley slice, though of a very different nature. We obtain applications concerning the general structure of the space of all two generator Kleinian groups and various universal constraints for Fuchsian groups.

Two-Generator Arithmetic Kleinian Groups II

We give here a pair of characterizations for a euclidean disk D which are concerned with the hyperbolic geometry in D and in domains which contain D.

KLEINIAN GROUPS WITH REAL PARAMETERS

The Kleinian group PGL(2, 03) is shown to have minimal covolume (, 0.0846 ... ) among all Kleinian groups containing torsion of order 6 (the associated hyperbolic orbifold is also the minimal volume cusped orbifold). This follows from: Any cocompact Kleinian group with torsion of order 6 has covolume at least 1 . As a consequence, any compact hyperbolic manifold with a symmetry of order 6 (with fixed points) has volume at least 4 . These results follow from new collaring theorems for torsion in a Kleinian group arising from our generalizations of the Shimizu-Leutbecher inequality.

Hyperbolic geometry and disks

A homeomorphism f of a set f of a set is said to be an L-quasiisometry of for all and if whenever ∞ ∈ E. Next f is said to have an M-quasiisometric extension to F ⊂ R 2 if there exists an M-quasiisometry g of E ∪ F such that g|E=f Suppose next that D is a Jordan domain in R 2 with boundary C and that ϕ is an L-quasiisometry of C. This paper contains complete solutions to the following Schoenflies type extension problems. Problem A. Given a domain D for which quasiisometries ϕ of C does ϕ have a quasiisometric extension g to D? Problem B. For which domains D does each quasiisometry ϕ of C have a quasiisometric extension g to D? It turns out that when C ⊂ R 2, each quasiisometry ϕ of C has a quasiisometric extension to D if and only if D is a quasidisk.