31 research outputs found
On a Generalization of the Hanoi Towers Group
In 2012, Bartholdi, Siegenthaler, and Zalesskii computed the rigid kernel for the only known group for which it is non-trivial, theHanoi towers group. There they determined the kernel was the Klein 4 group. We present a simpler proof of this theorem. In thecourse of the proof, we also compute the rigid stabilizers and present proofs that this group is a self-similar, self-replicating, regular branch group.
We then construct a family of groups which generalize the Hanoi towers group and study the congruence subgroup problem for the groups in this family. We show that unlike the Hanoi towers group, the groups in this generalization are just infinite and have trivial rigid kernel. We also put strict bounds on the branch kernel. Additionally, we show that these groups have subgroups of finite index with non-trivial rigid kernel, adding infinitely many new examples. Finally, we show that the topological closures of these groups have Hausdorff dimension arbitrarily close to 1
Simple groups separated by finiteness properties
We show that for every positive integer there exists a simple group that
is of type but not of type . For
these groups are the first known examples of this kind. They also provide
infinitely many quasi-isometry classes of finitely presented simple groups. The
only previously known infinite family of such classes, due to Caprace--R\'emy,
consists of non-affine Kac--Moody groups over finite fields. Our examples arise
from R\"over--Nekrashevych groups, and contain free abelian groups of infinite
rank.Comment: 25 pages. v2: incorporated comments v3: final version, to appear,
Invent. Mat
Finiteness properties for relatives of braided Higman--Thompson groups
We study the finiteness properties of the braided Higman--Thompson group
with labels in , and and
with labels in where is the braid group with strings and
is its pure braid subgroup. We show that for all and , the group (resp. or ) is of type
if and only if is. Our result in particular confirms a recent
conjecture of Aroca and Cumplido.Comment: 25 pages;first part of arXiv:2103.14589v1 with the second part to
appear separatel
Homological stability for the ribbon Higman--Thompson groups
We generalize the notion of asymptotic mapping class groups and allow them to
surject to the Higman--Thompson groups, answering a question of Aramayona and
Vlamis in the case of the Higman--Thompson groups. When the underlying surface
is a disk, these new asymptotic mapping class groups can be identified with the
ribbon and oriented ribbon Higman--Thompson groups. We use this model to prove
that the ribbon Higman--Thompson groups satisfy homological stability,
providing the first homological stability result for dense subgroups of big
mapping class groups. Our result can also be treated as an extension of
Szymik--Wahl's work on homological stability for the Higman--Thompson groups to
the surface setting.Comment: 23 pages; split off from arXiv:2103.14589v1 with the first part
available as arXiv:2103.14589v
Zipf ’s Law and Its Correlation to the GDP of Nations
This study looks at power laws, specifically Zipf ’s law and Pareto distributions, previously used to describe city size
distribution, income distribution within firms, and word distribution within languages and documents among other
things, and Gibrat’s law describing growth rate. This study seeks to discover if Zipf’s law can also be used to model the distribution of GDP’s worldwide using Gibrat’s law as a justification. The simplest method to determine Zipf's
law’s applicability, and the one used in this study, was to create a log log plot, plotting rank versus size of the GDPs. Using that plot, Zipf ’s law was verified through two criteria. First the plot must appear linear and second it must have a slope of -1. For the purpose of this study, the data looked at was for all countries and then countries split into categories of emerging economies and advanced economies for the years 2005, 2006, 2007, and 2008. The results of this study showed that all countries and countries with emerging economies did not appear linear on the log log plot while advanced economies appeared linear with a slope roughly -.70, suggesting that GDP distribution of advanced economies instead follow a Pareto distribution. Advanced economies also showed a significantly smaller variation in growth rates over the four years as implied by Gibrat’s law. This was used as a possible explanation for the distribution discovered
Block mapping class groups and their finiteness properties
A Cantor surface is a non-compact surface obtained by gluing
copies of a fixed compact surface (a block), with boundary
components, in a tree-like fashion. For a fixed subgroup , we
consider the subgroup whose elements
eventually send blocks to blocks and act like an element of ; we refer to
as the block mapping class group with local action
prescribed by . The family of groups so obtained contains the asymptotic
mapping class groups of \cite{SW21a,ABF+21, FK04}. Moreover, there is a natural
surjection onto the family symmetric Thompson groups of Farley--Hughes
\cite{FH15}; in particular, they provide a positive answer to \cite[Question
5.37]{AV20}. We prove that, when the block is a (holed) sphere or a (holed)
torus, is of type if and only if is of type .
As a consequence, for every , has a subgroup of type but
not which contains the mapping class group of every compact
subsurface of .Comment: 21 pages, 1 figur