Generalized Grigorchuk's Overgroups: Growth, Cluster Points, and Associated Rational Maps
Abstract
In this dissertation, we construct and investigate a family of groups, indexed by sequences of three symbols, that generalize the famous Grigorchuk's overgroup. Our work is spitted into three parts: (i) study of growth, (ii) study of topological and algebraic properties of the closure of this family in the space of marked 8-generated groups, and (iii) developing the technical tools of dynamic origin for study the spectral problems associated with the groups in this family.
In the first part, we show, if the sequence is eventually constant, then the corresponding group is of polynomial growth, and if the sequence is not eventually constant, then the group is of intermediate growth. In the case of non-eventually constant sequence, we give a universal lower bound for the growth rate and an upper bound for homogeneous sequences.
The second part contains the observation that this family is not closed, and the closure is the union of the (countable) set of isolated points and a Cantor set. The cluster points are constructed using branch-type algorithms and are closely related to the Lamplighter groups. Finally, we show that the generalized overgroups that are of intermediate growth are infinitely presented.
The final part is dedicated to studying the Schur complements and multi-dimensional rational maps associated with the generalized overgroups. First, we compute the Schur complements and multi-dimensional rational maps associated with some groups, including the generalized overgroups. These rational maps can be realized as two-dimensional and do belong to a two-parametric family of maps. The two-parametric maps have the integrability property of being semi-conjugate to the Chebyshev map. We show that any random iterations of two-parametric maps, viewed as maps on projective space, are algebraically stable in a rational variety.
Subject
Growth of groupsIntermediate growth
Grigorchuk group and Grigorchuk's overgroup
Grigorchuk’s space of marked groups
Word problem
Self-similar group
Rational map
Schur complement
Citation
Samarakoon, Supun Thamara (2021). Generalized Grigorchuk's Overgroups: Growth, Cluster Points, and Associated Rational Maps. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /195753.