Groups, Graphs and Trees

Abstract

Presenting groups in a formal, abstract algebraic manner is both useful and powerful, yet it avoids a fascinating geometric perspective on group theory - which is also useful and powerful, particularly in the study of infinite groups. This book presents the modern, geometric approach to group theory, in an accessible and engaging approach to the subject. Topics include group actions, the construction of Cayley graphs, and connections to formal language theory and geometry. Theorems are balanced by specific examples such as Baumslag-Solitar groups, the Lamplighter group and Thompson's group. Only exposure to undergraduate-level abstract algebra is presumed, and from that base the core techniques and theorems are developed and recent research is explored. Exercises and figures throughout the text encourage the development of geometric intuition. Ideal for advanced undergraduates looking to deepen their understanding of groups, this book will also be of interest to graduate students and researchers as a gentle introduction to geometric group theory.

The Planar Cayley Graphs are Effectively Enumerable I: Consistently Planar Graphs

“…We will follow the terminology of [3] for graph-theoretical terms and that of [1] and [17] for group-theoretical ones. Let us recall the definitions most relevant for this paper.…”

Isotropic quantum walks on lattices and the Weyl equation

“…Substituting into Eqs. (20), one derives α + = α − =: n and, up to a change of basis, A e = imV σ X with m ≥ 0. Imposing the normalization condition (7) amounts to the relation n 2 + m 2 = 1.…”

“…In Ref. [1] the derivation of the Weyl QWs exploited the technical assumption that there is a quasi-isometry [20] of the Cayley graph in a Euclidean manifold such that no vertex can lie within the * dariano@unipv.it † marco.erba01@ateneopv.it ‡ paolo.perinotti@unipv.it sphere of nearest neighbours. On the other hand, most of the derivation did not use the isotropy principle.…”

Quantum Cayley graphs for free groups

“…Let S ⊂ G be a finite generating set for G, meaning that every element of G can be expressed as a product of elements of S and their inverses. Following [17], the Cayley graph Γ G,S for the group G with generating set S is the directed graph whose vertex set V is the set of elements of G. The edge set of Γ G,S is the set E of directed edges (v, vs) with s ∈ S, initial vertex v ∈ G and terminal vertex vs. When confusion is unlikely we will simply write Γ for Γ G,S .…”

“…This work will focus on Cayley graphs with G = F M , the free group [17] with rank M . Recall that the elements of F M are equivalence classes of finite length words generated by M distinct symbols s 1 , .…”