Automorphisms of Groups and a Higher Rank JSJ Decomposition II: The single ended case

The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. In this sequence of papers we study the automorphisms of a hierarchically hyperbolic group (HHG) that satisfies some weak acylindricity conditions. To study these automorphisms we construct an object that can be viewed as a higher rank JSJ decomposition. This higher rank decomposition encodes the dynamics of individual automorphisms and the structure of the outer automorphism group of an HHG

Local Magnetization in the Boundary Ising Chain at Finite Temperature

We study the local magnetization in the 2-D Ising model at its critical temperature on a semi-infinite cylinder geometry, and with a nonzero magnetic field $h$ applied at the circular boundary of circumference $\beta$. This model is equivalent to the semi-infinite quantum critical 1-D transverse field Ising model at temperature $T \propto \beta^{-1}$, with a symmetry-breaking field $\propto h$ applied at the point boundary. Using conformal field theory methods we obtain the full scaling function for the local magnetization analytically in the continuum limit, thereby refining the previous results of Leclair, Lesage and Saleur in Ref. \onlinecite{Leclair}. The validity of our result as the continuum limit of the 1-D lattice model is confirmed numerically, exploiting a modified Jordan-Wigner representation. Applications of the result are discussed.Comment: 9 pages, 3 figure

More Than 1700 Years of Word Equations

Geometry and Diophantine equations have been ever-present in mathematics. Diophantus of Alexandria was born in the 3rd century (as far as we know), but a systematic mathematical study of word equations began only in the 20th century. So, the title of the present article does not seem to be justified at all. However, a linear Diophantine equation can be viewed as a special case of a system of word equations over a unary alphabet, and, more importantly, a word equation can be viewed as a special case of a Diophantine equation. Hence, the problem WordEquations: "Is a given word equation solvable?" is intimately related to Hilbert's 10th problem on the solvability of Diophantine equations. This became clear to the Russian school of mathematics at the latest in the mid 1960s, after which a systematic study of that relation began. Here, we review some recent developments which led to an amazingly simple decision procedure for WordEquations, and to the description of the set of all solutions as an EDT0L language.Comment: The paper will appear as an invited address in the LNCS proceedings of CAI 2015, Stuttgart, Germany, September 1 - 4, 201

Measurement of frequency occupancy levels in TV bands in urban environment in Kosovo

In this study we conduct an assessment of usage and availability of frequency bands, traditionally assigned to TV broadcasters, in urban environments in Kosovo. The assessment was performed for VHF and UHF bands at 8 different urban locations. Localized measurements indicate that a major part of these frequencies is severely under-utilized even in highly urbanized areas where a higher utilization level would be expected. Preliminary results further show that spectrum utilization level varies significantly with altitude and is much lower in indoor environments. Our initial calculations show that current percentage of availability of TV bands in tested locations varies between 87.5% and 100%. These results indicate that spectrum utilization in these bands could be greatly improved by allowing the opportunistic use of spectrum by cognitive radios and other wireless communication technologies, such as future cellular networks

No-splitting property and boundaries of random groups

We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1/2, are Menger curves.Comment: 20 page

Solution sets for equations over free groups are EDT0L languages

© World Scientific Publishing Company. We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jez, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to NSPACE(n log n) here

On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups

We show that the isomorphism problem is solvable in the class of central extensions of word-hyperbolic groups, and that the isomorphism problem for biautomatic groups reduces to that for biautomatic groups with finite centre. We describe an algorithm that, given an arbitrary finite presentation of an automatic group $\Gamma$, will construct explicit finite models for the skeleta of $K(\Gamma,1)$ and hence compute the integral homology and cohomology of $\Gamma$.Comment: 21 pages, 4 figure