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Anitha Thillaisundaram

Publications and source records attributed to Anitha Thillaisundaram.

At least 19 recordsLinked to original sources
arXiv2026

GGS-groups acting on trees of growing degrees

We consider analogues of Grigorchuk-Gupta-Sidki (GGS-)groups acting on trees of growing degree; the so-called growing GGS-groups. These groups are not just infinite and do not possess the congruence subgroup property, but many of them are branch and have the $p$-congruence subgroup property, for a prime $p$. Among them, we find groups with maximal subgroups only of finite index, and with infinitely many such maximal subgroups. These give the first examples of finitely generated branch groups with infinitely many finite-index maximal subgroups. Additionally, we prove that congruence quotients of growing GGS-groups associated to a defining vector of zero sum give rise to Beauville groups.

math.GR↗
arXiv2026

The Basilica group of the Grigorchuk-Erschler group

Francoeur and Garrido recently provided the first explicit examples of finitely generated branch groups with maximal subgroups of infinite index, where one of their examples is the Grigorchuk-Erschler group. Here, using the Basilica operation constructed by Petschick and Rajeev, we show that the second Basilica group of the Grigorchuk-Erschler group also has maximal subgroups of infinite index.

math.GR↗
arXiv2026

On the Frobenius number for three variables

For positive integers $a$, $b$, and $c$ which have no common divisor, the Frobenius number of $a$, $b$ and $c$ is defined to be the largest integer that cannot be expressed as a linear combination of $a$, $b$ and $c$ with non-negative integer coefficients. In 2017, Tripathi gave an algorithmic formula for the Frobenius number in three variables, however there were some minor inconsistencies in the formula. In this paper, we settle these inconsistencies.

math.NT↗
arXiv2025

The lower $p$-series of analytic pro-$p$ groups and Hausdorff dimension

Let $G$ be a $p$-adic analytic pro-$p$ group of dimension $d$. We produce an approximate series which descends regularly in strata and whose terms deviate from the lower $p$-series in a uniformly bounded way. This brings to light a new set of rational invariants, canonically associated to $G$, that yield the aforementioned uniform bound and that restrict the possible values for the Hausdorff dimensions of closed subgroups of $G$ with respect to the lower $p$-series. In particular, the Hausdorff spectrum of $G$ with respect to the lower $p$-series is discrete and consists of at most $2^d$ rational numbers.

math.GR↗
arXiv2025

Normal subgroups of non-torsion multi-EGS groups

We study the distribution of normal subgroups in non-torsion, regular branch multi-EGS groups and show that the congruence completions of such groups have bounded finite central width. In particular, we show that the profinite completion of the Fabrykowski--Gupta group acting on the $p$-adic tree has central width 2 for every odd prime $p$. The methods used also apply to the family of Sunic groups, which closely resemble the Grigorchuk group.

math.GR↗
arXiv2025

Invariable generation of certain branch groups

Let $G$ be a group. Then $S\subseteq G$ is an invariable generating set of $G$ if every subset $S'$ obtained from $S$ by replacing each element with a conjugate is also a generating set of $G$. We investigate invariable generation among key examples of branch groups. In particular, we prove that all generating sets of the torsion Grigorchuk groups, of the branch Grigorchuk-Gupta-Sidki groups and of the torsion multi-EGS groups (which are natural generalisations of the Grigorchuk-Gupta-Sidki groups) are invariable generating sets. Furthermore, for the first Grigorchuk group and the torsion Grigorchuk-Gupta-Sidki groups, every finitely generated subgroup has a finite invariable generating set. Our results apply to finitely generated groups in $\mathcal{MN}$, the class of groups whose maximal subgroups are all normal. We then obtain that any $2$-generated group in $\mathcal{MN}$ is almost $\frac{3}{2}$-generated, and end by applying this observation to generating graphs.

math.GR↗
arXiv2025

On the Hausdorff spectra of free pro-$p$ groups and certain $p$-adic analytic groups

We establish that finitely generated non-abelian direct products $G$ of free pro-$p$ groups have full Hausdorff spectrum with respect to the lower $p$-series $\mathcal{L}$. This complements similar results with respect to other standard filtration series and a recent theorem showing that the Hausdorff spectrum $\text{hspec}^\mathcal{L}(G)$ of a $p$-adic analytic pro-$p$ group $G$ is discrete and consists of at most $2^{\dim(G)}$ rational numbers. The latter also left some room for improvement regarding the upper bound. Indeed, for finitely generated nilpotent pro-$p$ groups $G$ we obtain the stronger assertion that the cardinality of the Hausdorff spectrum is at most the analytic dimension of $G$. Moreover, we produce a corresponding result when the $p$-adic analytic pro-$p$ group $G$ is just infinite, which holds not just for the lower $p$-series but for arbitrary filtration series. Finally, we show that, if $G$ is a countably based pro-$p$ group with an open subgroup mapping onto the free abelian pro-$p$ group $\mathbb{Z}_p \oplus \mathbb{Z}_p$, then for every prescribed finite set $\{0,1\} \subseteq X \subseteq [0,1]$ there is a filtration series $\mathcal{S}$ such that $\text{hspec}^\mathcal{S}(G) = X$; in particular, $|\text{hspec}^{\mathcal{S}}(G)|$ is unbounded, as $\mathcal{S}$ runs through all filtration series of $G$ with $|\text{hspec}^{\mathcal{S}}(G)| < \infty$.

math.GR↗
arXiv2024

Maximal subgroups of a family of iterated monodromy groups

The Basilica group is a well-known 2-generated weakly branch, but not branch, group acting on the binary rooted tree. Recently a more general form of the Basilica group has been investigated by Petschick and Rajeev, which is an $s$-generated weakly branch, but not branch, group that acts on the $m$-adic tree, for $s,m > 1$. A larger family of groups, which contains these generalised Basilica groups, is the family of iterated monodromy groups. With the new developments by Francoeur, the study of the existence of maximal subgroups of infinite index has been extended from branch groups to weakly branch groups. Here we show that a subfamily of iterated monodromy groups, which more closely resemble the generalised Basilica groups, have maximal subgroups only of finite index.

math.GR↗
arXiv2024

Profinite groups with soluble centralisers

We show that a profinite group, in which the centralisers of non-trivial elements are metabelian, is either virtually pro-$p$ or virtually soluble of derived length at most 4. We furthermore show that a prosoluble group, in which the centralisers of non-trivial elements are soluble of bounded derived length, is either soluble or virtually pro-$p$.

math.GR↗
arXiv2023

Beauville structures for quotients of generalised GGS-groups

A finite group with a Beauville structure gives rise to a certain compact complex surface called a Beauville surface. Gül and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki (GGS-)groups that act on the $p$-adic tree, for $p$ an odd prime, admit Beauville structures. We extend their result by showing that quotients of infinite periodic GGS-groups acting on $p^n$-adic trees, for $p$ any prime and $n\ge 2$, also admit Beauville structures.

math.GR↗
arXiv2023

The Amit-Ashurst conjecture for finite metacyclic p-groups

The Amit conjecture about word maps on finite nilpotent groups has been shown to hold for certain classes of groups. The generalised Amit conjecture says that the probability of an element occurring in the image of a word map on a finite nilpotent group G is either 0, or at least 1/|G|. Noting the work of Ashurst, we name the generalised Amit conjecture the Amit-Ashurst conjecture and show that the Amit-Ashurst conjecture holds for finite p-groups with a cyclic maximal subgroup.

math.GR↗
arXiv2022

Ramification structures for quotients of multi-EGS groups

Groups associated to surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called ramification structure. Gül and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, admit ramification structures. We extend their result by showing that quotients of generalisations of the GGS-groups, namely multi-EGS groups, also admit ramification structures.

math.GR↗
arXiv2022

Conjugacy classes of polyspinal groups

Spinal groups and multi-GGS groups are both generalisations of the well-known Grigorchuk-Gupta-Sidki (GGS-)groups. Here we give a necessary condition for spinal groups to be conjugate, and we establish a necessary and sufficient condition for multi-GGS groups to be conjugate. We also introduce a natural common generalisation of both classes, which we call polyspinal groups. Our results enable us to give a negative answer to a question of Bartholdi, Grigorchuk and Sunik, on whether every finitely generated branch group is isomorphic to a weakly branch spinal group.

math.GR↗
arXiv2022

A pro-2 group with full normal Hausdorff spectra

We construct a $2$-generated pro-$2$ group with full normal Hausdorff spectrum $[0,1]$, with respect to each of the four standard filtration series: the $2$-power series, the lower $2$-series, the Frattini series, and the dimension subgroup series. This answers a question of Klopsch and the second author, for the even prime case; the odd prime case was settled by the first author and Klopsch. Also, our construction gives the first example of a finitely generated pro-$2$ group with full Hausdorff spectrum with respect to the lower $2$-series.

math.GR↗
arXiv2021

Maximal subgroups of non-torsion Grigorchuk-Gupta-Sidki groups

A Grigorchuk-Gupta-Sidki (GGS-)group is a subgroup of the automorphism group of the $p$-adic tree for an odd prime $p$, generated by one rooted automorphism and one directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups of infinite index. Here we extend the result to non-torsion GGS-groups, which include the weakly regular branch, but not branch, GGS-group.

math.GR↗
arXiv2021

Ramification structures for quotients of the Grigorchuk groups

Groups associated to surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called ramification structure. In this paper, we prove that infinitely many quotients of the Grigorchuk groups admit ramification structures. This gives the first explicit infinite family of 3-generated finite 2-groups with ramification structures.

math.GR↗
arXiv2021

The finitely generated Hausdorff spectra of a family of pro-$p$ groups

Recently the first example of a family of pro-$p$ groups, for $p$ a prime, with full normal Hausdorff spectrum was constructed. In this paper we further investigate this family by computing their finitely generated Hausdorff spectrum with respect to each of the five standard filtration series: the $p$-power series, the iterated $p$-power series, the lower $p$-series, the Frattini series and the dimension subgroup series. Here the finitely generated Hausdorff spectra of these groups consist of infinitely many rational numbers, and their computation requires a rather technical approach. This result also gives further evidence to the non-existence of a finitely generated pro-$p$ group with uncountable finitely generated Hausdorff spectrum.

math.GR↗