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 dimension of their closures, providing...

A. We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. We apply this method to construct a forcing (without using an inaccessible or amalgamation) that makes all definable sets of reals measurable with respect to a certain (non-ccc) ideal. 1. I In the seminal paper [10] Solovay proved that in the Levy model (after collap...

We study automorphisms of a relatively hyperbolic group G. When G is oneended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GLn(Z) fixing certain basis elements. When more general parabolic groups are allowed, these ...

In this paper we prove that a vertex-centered automorphism of a tree gives a proper factor of the characteristic polynomial of its distance or adjacency matrix. We also show that the characteristic polynomial of the distance matrix of any graph always has a factor of degree equal to the number of vertex orbits of the graph. These results are applied to full k-ary trees and some other problems. ...

in this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. with this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. n(g) and s(g) are the set of all nilpotency classes and the set of all solvability classes for the group g with respect to different automorphi...

Let k be a field. The Cremona group Bir(Pk) of k in dimension d is defined as the group of birational transformations of the d-dimensional k-affine space. It can also be described as the group of k-automorphisms of the field of rational functions k(t1, . . . , td). We endow it with the discrete topology. Let us say that a group has Property (FR)∞ if it satisfies the following (1) For every isom...

We study various properties of a random graph Rn, drawn uniformly at random from the class An of all simple graphs on n labelled vertices that satisfy some given property, such as being planar or having tree-width at most k. In particular, we show that if the class A is ‘small’ and ‘addable’, then the probability that Rn is connected is bounded away from 0 and from 1. As well as connectivity we...