We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior. The free monoid X * of words… (More)
It is shown that certain ascending HNN extensions of free abelian groups of finite rank, as well as various lamplighter groups, can be realized as automaton groups, i.e., can be given a self-similar structure. This includes the solvable Baumslag-Solitar groups BS(1, m), for m = ±1. In addition, it is shown that, for any relatively prime integers m, n ≥ 2,… (More)
We generalize a class of groups defined by Rostislav Grigorchuk in [Gri84] to a much larger class of groups, and provide upper and lower bounds for their word growth (they are all of intermediate growth) and period growth (under a small additional condition, they are periodic).
The notion of upper distortion for graded submonoids embedded in groups and monoids is introduced. A finitely generated monoid M is graded if every element of M can be written in only finitely many ways in terms of some fixed system of generators. Examples of such monoids are free monoids, Artin monoids, and monoids satisfying certain small cancellation… (More)
We introduce L-presentations: group presentations endowed with a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite L-presentations.
We introduce a transformation of finite integer sequences, show that every sequence eventually stabilizes under this transformation and that the number of fixed points is counted by the Catalan numbers. The sequences that are fixed are precisely those that describe themselves — every term t is equal to the number of previous terms that are smaller than t.… (More)
We show that the bandwidth of a square two-dimensional grid of arbitrary size can be reduced if two (but not less than two) edges are deleted. The two deleted edges may not be chosen arbitrarily, but they may be chosen to share a common endpoint or to be non-adjacent. We also show that the bandwidth of the rectangular n × m (m ≥ n) grid can be reduced by k,… (More)
We introduce a transformation on integer sequences for which the set of images is in bijective correspondence with the set of Young tableaux. We use this correspondence to show that the set of images, known as ballot sequences, is also the set of double points of our transformation. In the second part, we introduce other transformations of integer sequences… (More)