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Homomorphisms to constructed from random walks

Anna Erschler, Anders Karlsson (2010)

Annales de l’institut Fourier

We give a construction of homomorphisms from a group into the reals using random walks on the group. The construction is an alternative to an earlier construction that works in more general situations. Applications include an estimate on the drift of random walks on groups of subexponential growth admitting no nontrivial homomorphism to the integers and inequalities between the asymptotic drift and the asymptotic entropy. Some of the entropy estimates obtained have applications independent of the...

Homotopy types of one-dimensional Peano continua

Katsuya Eda (2010)

Fundamenta Mathematicae

Let X and Y be one-dimensional Peano continua. If the fundamental groups of X and Y are isomorphic, then X and Y are homotopy equivalent. Every homomorphism from the fundamental group of X to that of Y is a composition of a homomorphism induced from a continuous map and a base point change isomorphism.

Horocyclic products of trees

Laurent Bartholdi, Markus Neuhauser, Wolfgang Woess (2008)

Journal of the European Mathematical Society

Let T 1 , , T d be homogeneous trees with degrees q 1 + 1 , , q d + 1 3 , respectively. For each tree, let 𝔥 : T j be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of T 1 , , T d is the graph 𝖣𝖫 ( q 1 , , q d ) consisting of all d -tuples x 1 x d T 1 × × T d with 𝔥 ( x 1 ) + + 𝔥 ( x d ) = 0 , equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If d = 2 and q 1 = q 2 = q then we obtain a Cayley graph of the...

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