A characterization of the locally finite networks admitting non-constant harmonic functions of finite energy

We characterize the locally finite networks admitting non-constant harmonic functions of finite energy. Our characterization unifies the necessary existence criteria of Thomassen and of Lyons and Peres with the sufficient criterion of Soardi. We also extend a necessary existence criterion for non-elusive non-constant harmonic functions of finite energy due to Georgakopoulos

Elusive Codes in Hamming Graphs

We consider a code to be a subset of the vertex set of a Hamming graph. We examine elusive pairs, code-group pairs where the code is not determined by knowledge of its set of neighbours. We construct a new infinite family of elusive pairs, where the group in question acts transitively on the set of neighbours of the code. In our examples, we find that the alphabet size always divides the length of the code, and prove that there is no elusive pair for the smallest set of parameters for which this is not the case. We also pose several questions regarding elusive pairs

Deciding Isomorphy using Dehn fillings, the splitting case

We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if the graphs of groups have isomorphic vertex groups and, if so, whether a global isomorphism can be assembled. For the class of finitely generated nilpotent groups, we give solutions to these algorithmic problems by using the arithmetic nature of these groups and of their automorphism groups.Comment: 76 pages. This version incorporates referee comments and corrections. The main changes to the previous version are a better treatment of the algorithmic recognition and presentation of virtually cyclic subgroups and a new proof of a rigidity criterion obtained by passing to a torsion-free finite index subgroup. The previous proof relied on an incorrect result. To appear in Inventiones Mathematica

Metric structures in L_1: Dimension, snowflakes, and average distortion

We study the metric properties of finite subsets of L_1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L_1.Comment: 9 pages, 1 figure. To appear in European Journal of Combinatorics. Preliminary version appeared in LATIN '0