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This note records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group G has an element whose normal closure is abelian and of finite index, then G has a finite generating set(More)
We study the sofic tree shifts of A ∗ , where Σ∗ is a regular rooted tree of finite rank. In particular, we give their characterization in terms of unrestricted Rabin automata. We show that if X ⊂ A ∗ is a sofic tree shift, then the configurations in X whose orbit under the shift action is finite are dense in X , and, as a consequence of this, we deduce(More)
A connection relating Tamari lattices on symmetric groups regarded as lattices under the weak Bruhat order to the positive monoid P of Thompson group F is presented. Tamari congruence classes correspond to classes of equivalent elements in P . The two well known normal forms in P correspond to endpoints of intervals in the weak Bruhat order that determine(More)
For each prime p and a monic polynomial f , invertible over p, we define a group Gp,f of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group G2,x2+x+1. We show that the constructed groups are self-similar, regular branch groups. This enables us to calculate the Hausdorff(More)
We provide a self-similar measure for the self-similar group G acting faithfully on the binary rooted tree, defined as the iterated monodromy group of the quadratic polynomial z + i. We also provide an Lpresentation for G and calculations related to the spectrum of the Markov operator on the Schreier graph of the action of G on the orbit of a point on the(More)
Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the quotient of some group by its center) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a(More)
In the Twin Towers of Hanoi version of the well known Towers of Hanoi Problem there are two coupled sets of pegs. In each move, one chooses a pair of pegs in one of the sets and performs the only possible legal transfer of a disk between the chosen pegs (the smallest disk from one of the pegs is moved to the other peg), but also, simultaneously, between the(More)