We establish that a quasiconformal group is of compact type if and only if its limits set is purely conical and find that the limit set of a quasiconformal group of compact type is uniformly perfect. A key tool is the result of Bowditch-Tukia on compact-type convergence groups. These results provide crucial tools for studying the deformations of quasiconformal groups and in establishing isomorphisms between such groups and conformal groups.

We prove that a K-quasiconformal mapping f:ℝ² → ℝ² which maps the unit disk onto itself preserves the space EXP() of exponentially integrable functions over , in the sense that u ∈ EXP() if and only if $u\circ f^{-1}\in EXP\left(\right)$. Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate $1/(1+KlogK)\le \left(\right||u\circ f^{-1}{\left|\right|}_{EXP\left(\right)}{)/(\left|\right|u\left|\right|}_{EXP\left(\right)})\le 1+KlogK$ for every u ∈ EXP(). Similarly, we consider the distance from $L^{\infty }$ in EXP and we prove that if f: Ω → Ω’ is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then $1/K\le \left(dis{t}_{EXP\left(f\right(G\left)\right)}(u\circ f^{-1},L^{\infty }\left(f\left(G\right)\right))\right)/\left(dis{t}_{EXP\left(f\right(G\left)\right)}(u,L^{\infty }\left(G\right))\right)\le K$ for every u ∈ EXP(). We also prove that...

We examine how Poincaré change under quasiconformal maps between appropriate metric spaces having the same Hausdorff dimension. We also show that for many metric spaces the Sobolev functions can be identified with functions satisfying Poincaré, and this allows us to extend to the metric space setting the fact that quasiconformal maps from ${ℝ}^Q$ onto ${ℝ}^Q$ preserve the Sobolev space $L^{1,Q}\left({ℝ}^Q\right)$.

In this paper we construct quasiconformal mappings between Y-pieces so that the corresponding Beltrami coefficient has exponential decay away from the boundary. These maps are used in a companion paper to construct quasiFuchsian groups whose limit sets are non-rectifiable curves of dimension 1.

In this paper we study quasiconformal homeomorphisms of the unit ball B = Bn = {x ∈ Rn: |x| < 1} of Rn onto John domains. We recall that John domains were introduced by F. John in his study of rigidity of local quasi-isometries [J]; the term John domain was coined by O. Martio and J. Sarvas seventeen years later [MS]. From the various equivalent characterizations we shall adapt the following definition based on diameter carrots, cf. [V4], [V5], [NV].

We consider quasiconformal mappings in the upper half space ${ℝ}_{+}^{n+1}$ of ${ℝ}^{n+1}$, $n\ge 2$, whose almost everywhere defined trace in ${ℝ}^n$ has distributional differential in $L^n\left({ℝ}^n\right)$. We give both geometric and analytic characterizations for this possibility, resembling the situation in the classical Hardy space $H^1$. More generally, we consider certain positive functions defined on ${ℝ}_{+}^{n+1}$, called conformal densities. These densities mimic the averaged derivatives of quasiconformal mappings, and we prove analogous trace theorems for them....

We study the behaviour of a quasiconformal mapping when we change the norms of the considered normed spaces by other equivalent norms. We propose a new metric definition with which we can study the interdependence between a quasiconformal homeomorphism and the new equivalent norms of the normed spaces.