Bootstrapping in convergence groups

We prove a true bootstrapping result for convergence groups acting on a Peano continuum. We give an example of a Kleinian group H which is the amalgamation of two closed hyperbolic surface groups along a simple closed curve. The limit set Lambda H is the closure of a `tree of circles' (adjacent circles meeting in pairs of points). We alter the action of H on its limit set such that H no longer acts as a convergence group, but the stabilizers of the circles remain unchanged, as does the action of a circle stabilizer on said circle. This is done by first separating the circles and then gluing them together backwards.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-32.abs.htm

Convergence groups from subgroups

We give sufficient conditions for a group of homeomorphisms of a Peano continuum X without cut-points to be a convergence group. The condition is that there is a collection of convergence subgroups whose limit sets `cut up' X in the correct fashion. This is closely related to the result in [E Swenson, Axial pairs and convergence groups on S^1, Topology 39 (2000) 229-237].Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper22.abs.htm

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

Combination of convergence groups

We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an acylindrical amalgamation of relatively hyperbolic groups over a fully quasi-convex subgroup. We apply our result to Sela's theory on limit groups and prove their relative hyperbolicity. We also get a proof of the Howson property for limit groups.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper27.abs.htm

Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary

Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper7.abs.htm

Effectively Closed Infinite-Genus Surfaces and the String Coupling

The class of effectively closed infinite-genus surfaces, defining the completion of the domain of string perturbation theory, can be included in the category $O_G$, which is characterized by the vanishing capacity of the ideal boundary. The cardinality of the maximal set of endpoints is shown to be 2^{\mit N}. The product of the coefficient of the genus-g superstring amplitude in four dimensions by $2^g$ in the $g\to \infty$ limit is an exponential function of the genus with a base comparable in magnitude to the unified gauge coupling. The value of the string coupling is consistent with the characteristics of configurations which provide a dominant contribution to a finite vacuum amplitude.Comment: TeX, 33 page

CAT(0) groups with specified boundary

We specify exactly which groups can act geometrically on CAT(0) spaces whose visual boundary is homeomorphic to either a circle or a suspension of a Cantor set.Comment: This is the version published by Algebraic & Geometric Topology on 4 May 200

Measurement and Evaluation of IP Multimedia Subsystem

Tämän diplomityön tarkoituksena oli tutkia IP-multimedia-alijärjestelmän (IMS) suorituskykyä. IMS on IETF:n, erityisesti SIP ja Diameter, protokolliin perustuva arkkitehtuurimalli, jota ollaan ottamassa käyttöön seuraavan sukupolven 4Gverkoissa. Työn kirjallisessa osuudessa esitellään IMS-arkkitehtuurimalli, kuten sen on määritellyt pääasiassa 3GPP, sekä käytettävät käyttötapaukset, jotka TISPAN on määritellyt standardissaan IMS-suorituskykymittauksista. Työn empiirisessä osassa esitellään suorituskykymittausten tuloksia tutkimusympäristöön rakennetusta IMS-verkosta ja vertailukohtana käytetystä SIP-verkosta. Suorituskykymittauksissa tehtiin alustavat mittaukset IMS-verkossa, joiden tulosten perusteella pystyttiin parametrisoimaan asiallisesti varsinaiset mittaukset. Varsinaiset mittaukset tehtiin samoilla parametreilla IMS ja SIP -verkoissa, kummassakin verkossa kahdessa eri mittaushaarassa - rekisteröitymisiin liittyvät mittaukset ja puheluihin ja viestinvälityksiin liittyvät mittaukset. Näissä mittauksissa mitattiin viiteen eri skenaarioon liittyvien merkinantojen viiveitä ja kestoja. Nämä skenaariotyypit olivat onnistunut rekisteröityminen, uudelleenrekisteröityminen, rekisteröitymisen lopettaminen, onnistunut puhelu ja onnistunut viestinvälitys. Mittausten lisäksi pohdittiin eri skenaarioiden teoreettisia maksimikapasiteetteja. Saatujen mittaustulosten perusteella voidaan todeta, että tällä hetkellä puhdas SIP-verkko on IMS-verkkoa suorituskykyisempi, niin käytännön toteutuksessa kuin teoreettisesti lasketuissa maksimikapasiteettiarvioissakin.The purpose of this master's thesis was to study the performance of the IP Multimedia Subsystem (IMS). IMS is an architectual model based on IETF's protocols, especially SIP and Diameter, that will be introduced with the next generation 4G networks. The literature study of this thesis introduces the IMS architectual model as it has been defined mostly by 3GPP. The literature study introduces the use case parts of the standard of IMS/NGN Performance Benchmark defined by TISPAN. The empirical part of this thesis presents the results of the performance measurements of the IMS and the SIP networks in a research network. Before the actual performance measurements preliminary measurements were performed to be able to properly parametrize the actual performance measurements. The actual performance measurements were performed with the same parameters in the IMS and the SIP networks, in two branches on both networks - registration related measurements and calls and messaging related measurements. In the measurements we measured delays and durations of the signalling in five different type of scenarios. These scenario types were registration, re-registration, de-registration, successful call and successful messaging. We also considered the theoretical maximum performance limits for the scenarios. Based on the given measurement results we can show that the pure SIP network is more efficient than the IMS network in practice and also in the theoretical calculated maximum performance limits

Nonpositively curved 2-complexes with isolated flats

We introduce the class of nonpositively curved 2-complexes with the Isolated Flats Property. These 2-complexes are, in a sense, hyperbolic relative to their flats. More precisely, we show that several important properties of Gromov-hyperbolic spaces hold `relative to flats' in nonpositively curved 2-complexes with the Isolated Flats Property. We introduce the Relatively Thin Triangle Property, which states roughly that the fat part of a geodesic triangle lies near a single flat. We also introduce the Relative Fellow Traveller Property, which states that pairs of quasigeodesics with common endpoints fellow travel relative to flats, in a suitable sense. The main result of this paper states that in the setting of CAT(0) 2-complexes, the Isolated Flats Property is equivalent to the Relatively Thin Triangle Property and is also equivalent to the Relative Fellow Traveller Property.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper5.abs.htm

Kleinian groups and the rank problem

We prove that the rank problem is decidable in the class of torsion-free word-hyperbolic Kleinian groups. We also show that every group in this class has only finitely many Nielsen equivalence classes of generating sets of a given cardinality.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper12.abs.htm