Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Roughly speaking, a variety of groups is a class of groups in which certain relations called laws or rules or identities are universally valid. The most widely investigated variety of groups is that of abelian groups in which the commutative law is universally valid or, in other words, in which for arbitrary elements x1, x2 of the group, the identity 

$$ \left( {{x_1},{x_2}} \right) = 1 $$

holds, where, as usual, (x1, x2) is defined as the commutator 

$$ x_1^{ - 1}x_2^{ - 1}{x_1}{x_2} $$

of x1 and x2. Similarly, the Burnside variety of exponent e is defined as the class of groups in which, for all elements x, the relation 

$$ {x^e} = 1 $$

holds. The metabelian groups of Chapter II.6 can be defined as the variety of groups define4 by the identity 

$$ \left( {\left( {{x_1},{x_2}} \right),\left( {{x_3},{x_4}} \right)} \right) = 1, $$

and the nilpotent groups of class c discussed in Chapter II.7 form the variety defined by the identity 

$$ \left( {{x_1},{x_2},...,{x_c},{x_{c + 1}}} \right) = 1 $$

where the parentheses ( ) denote the simple (c + l)-fold commutator defined in Chapter II.7.

Varieties of Groups

For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those factors, which commute with all other factors, and the existential theories of the remaining factors. Both results also include suitable constraints for the variables. Larger classes of constraints lead in many cases to undecidability results.

Word equations over graph products

It is shown that the existential theory of ${\mathbb G}$ with rational constraints, over an HNN-extension ${\mathbb G}=\langle {\mathbb H},t; t^{-1}at=\varphi(a) (a \in A) \rangle$ is decidable, provided that the same problem is decidable in the base group ${\mathbb H}$ and that A is a finite group. The positive theory of ${\mathbb G}$ is decidable, provided that the existential positive theory of ${\mathbb G}$ is decidable and that A and ϕ(A) are proper subgroups of the base group ${\mathbb H}$ with A ∩ϕ(A) finite. Analogous results are also shown for amalgamated products. As a corollary, the positive theory and the existential theory with rational constraints of any finitely generated virtually-free group is decidable

Theories of HNN-Extensions and amalgamated products

This is the first paper in a series of three where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Our main goal is to show that group actions on Z^n-trees give one a powerful tool to study groups. All finitely generated groups acting freely on -trees also act freely on some Z^n-trees, but the latter ones form a much larger class. The natural effectiveness of all constructions for $Z^n-actions (which is not the case for R-trees) comes along with a robust algorithmic theory. In this paper we describe the algebraic structure of finitely generated groups acting freely and regularly on Z^n-trees and give necessary and sufficient conditions for such actions.

Groups with free regular length functions in Z^n

In this paper we survey recent developments in the theory of groups acting on Λ-trees. We are trying to unify all significant methods and techniques, both classical and recently developed, in an attempt to present various faces of the theory and to show how these methods can be used to solve major problems about finitely presented Λ-free groups. Besides surveying results known up to date we draw many new corollaries concerning structural and algorithmic properties of such groups.

Actions, length functions, and non-archimedean words

In the present paper we continue studying regular free group actions on Z^n-trees. We show that every finitely generated Z^n-free group G can be embedded into a finitely generated Z^n-free group H acting regularly on the underlying Z^n-tree (we call H a regular Z^n-completion of G) so that the action of G is preserved. Moreover, if G is effectively represented as a group of Z^n-words then the construction of H is effective and H is effectively represented as a group of Z^n-words.

Regular completions of Z^n-free groups

Using graph-theoretic techniques for f.g. subgroups of $F^{\mathbb{Z}[t]}$ we provide a criterion for a f.g. subgroup of a f.g. fully residually free group to be of finite index. Moreover, we show that this criterion can be checked effectively. Also we obtain an analogue of Greenberg-Stallings Theorem for f.g. fully residually free groups, and prove that a f.g. non-abelian subgroup of a f.g. fully residually free group is of finite index in its commensurator.

Finite index subgroups of fully residually free groups

In this paper we study random walks on a finitely generated group $G$ which has a free action on a $\mathbb{Z}^n$-tree. We show that if $G$ is non-abelian and acts minimally, freely and without inversions on a locally finite $\mathbb{Z}^n$-tree $\Gamma$ with the set of open ends ${\rm Ends}(\Gamma)$, then for every non-degenerate probability measure $\mu$ on $G$ there exists a unique $\mu$-stationary probability measure $\nu_\mu$ on ${\rm Ends}(\Gamma)$, and the space $({\rm Ends}(\Gamma), \nu_\mu)$ is a $\mu$-boundary. Moreover, if $\mu$ has finite first moment with respect to the word metric on $G$ (induced by a finite generating set), then the measure space $({\rm Ends}(\Gamma), \nu_\mu)$ is isomorphic to the Poisson--Furstenberg boundary of $(G, \mu)$.

Boundaries of Z^n-free groups

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