Rings and bilipschitz maps in normed spaces

We define the geometric modulus GM(A) of a ring A in a normed space And show that a set-bounded homeomorphism f: E→E is bilipschitz if and only if ⌈GM(A)-GM(fA)⌉ ≤ c for all rings A⊂E.Peer reviewe

Definitions of quasiconformality

We establish that the infinitesimal “ H -definition” for quasiconformal mappings on Carnot groups implies global quasisymmetry, and hence the absolute continuity on almost all lines. Our method is new even in R n where we obtain that the “limsup” condition in the H -definition can be replaced by a “liminf” condition. This leads to a new removability result for (quasi)conformal mappings in Euclidean spaces. An application to parametrizations of chord-arc surfaces is also given.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46582/1/222_2005_Article_BF01241122.pd

On torsionless subgroups of infinitely generated Fuchsian groups

A very useful property of finitely generated Fuchsian groups is that such a group always contains a normal subgroup of finite index with no elliptic elements. In this paper we prove an analogous result for infinitely generated Fuchsian groups. Obviously, in this case rwe cannot always find a torsionless subgroup of finite inderx but it turns out that a very pleasant analogy exists. We also determine when there exists a torsionless normal subgroup of finite index. Let G be a finitely generated Fuchsian group. Essentially, all proofs of the fact that G contains a normal subgroup of finite index without elliptic elements use the following method. One can find a finite group Fand a homomorphism E: G*F such that q-t(e)cG is torsionless (e is the neutral element of F). Then N:E-'(e) is necessarily of finite index in G and does not contain elliptic elements. ln 12, 3, 5l and [9, Chapter IV.10] the proof was based on a description of G in terms of gen-erators and relations. In contrast, Selberg [6] considered G as a subgroup of the group of real or complex nXn mattices. Thus the theorem is, in fact, true for all finitely generated matrix groups, and also for Kleinian groups ' Originally, th

On torsionless subgroups of infinitely generated Fuchsian groups

A very useful property of finitely generated Fuchsian groups is that such a group always contains a normal subgroup of finite index with no elliptic elements. In this paper we prove an analogous result for infinitely generated Fuchsian groups. Obviously, in this case rwe cannot always find a torsionless subgroup of finite inderx but it turns out that a very pleasant analogy exists. We also determine when there exists a torsionless normal subgroup of finite index. Let G be a finitely generated Fuchsian group. Essentially, all proofs of the fact that G contains a normal subgroup of finite index without elliptic elements use the following method. One can find a finite group Fand a homomorphism E: G*F such that q-t(e)cG is torsionless (e is the neutral element of F). Then N:E-'(e) is necessarily of finite index in G and does not contain elliptic elements. ln 12, 3, 5l and [9, Chapter IV.10] the proof was based on a description of G in terms of gen-erators and relations. In contrast, Selberg [6] considered G as a subgroup of the group of real or complex nXn mattices. Thus the theorem is, in fact, true for all finitely generated matrix groups, and also for Kleinian groups ' Originally, th