Cycle decompositions: from graphs to continua

We generalise a fundamental graph-theoretical fact, stating that every element of the cycle space of a graph is a sum of edge-disjoint cycles, to arbitrary continua. To achieve this we replace graph cycles by topological circles, and replace the cycle space of a graph by a new homology group for continua which is a quotient of the first singular homology group $H_1$. This homology seems to be particularly apt for studying spaces with infinitely generated $H_1$, e.g. infinite graphs or fractals.Comment: Advances in Mathematics (2011

On planar Cayley graphs and Kleinian groups

Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X \subseteq \mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ by another surface $Y \subseteq \mathbb{S}^2$. We also prove that if a group $H$ has a finitely generated Cayley (multi-)graph $C$ covariantly embeddable in $\mathbb{S}^2$, then $C$ can be chosen so as to have no infinite path on the boundary of a face. The proofs of these facts are intertwined, and the classes of groups they define coincide. In the orientation-preserving case they are exactly the (isomorphism types of) finitely generated Kleinian function groups. We construct a finitely generated planar Cayley graph whose group is not in this class. In passing, we observe that the Freudenthal compactification of every planar surface is homeomorphic to the sphere

Lamplighter graphs do not admit harmonic functions of finite energy

We prove that a lamplighter graph of a locally finite graph over a finite graph does not admit a non-constant harmonic function of finite Dirichlet energy

On graph-like continua of finite length

We extend the notion of effective resistance to metric spaces that are similar to graphs but can also be similar to fractals. Combined with other basic facts proved in the paper, this lays the ground for a construction of Brownian Motion on such spaces completed in [10]