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, calledâ€¦ (More)

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 Catalanâ€¦ (More)

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 manyâ€¦ (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-similarâ€¦ (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 intermediateâ€¦ (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 explicitlyâ€¦ (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, butâ€¦ (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 ofâ€¦ (More)