arXiv subjects

Anitha Thillaisundaram

Publications and source records attributed to Anitha Thillaisundaram.

32 records · Page 2Linked to original sources

arXiv2021

$p$-Basilica groups

We consider a generalisation of the Basilica group to all odd primes: the $p$-Basilica groups acting on the $p$-adic tree. We show that the $p$-Basilica groups have the $p$-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the $p$-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the Hausdorff dimensions of the $p$-Basilica groups. Lastly, we show that the $p$-Basilica groups do not possess maximal subgroups of infinite index and that they have infinitely many non-normal maximal subgroups.

math.GR↗

arXiv2020

The profinite completion of multi-EGS groups

The class of multi-EGS groups is a generalisation of the well-known Grigorchuk-Gupta-Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally our results show that branch multi-EGS groups are just infinite.

math.GR↗

arXiv2020

Hausdorff dimension of the second Grigorchuk group

We show that the Hausdorff dimension of the closure of the second Grigorchuk group is 43/128. Furthermore we establish that the second Grigorchuk group is super strongly fractal and that its automorphism group equals its normaliser in the full automorphism group of the tree.

math.GR↗

arXiv2020

Word problems for finite nilpotent groups

Let $w$ be a word in $k$ variables. For a finite nilpotent group $G$, a conjecture of Amit states that $N_w(1) \ge |G|^{k-1}$, where $N_w(1)$ is the number of $k$-tuples $(g_1,...,g_k)\in G^{(k)}$ such that $w(g_1,...,g_k)=1$. Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit's conjecture, and prove that $N_w(g) \ge |G|^{k-2}$, where $g$ is a $w$-value in $G$, for finite groups $G$ of odd order and nilpotency class 2. If $w$ is a word in two variables, we further show that $N_w(g) \ge |G|$, where $g$ is a $w$-value in $G$ for finite groups $G$ of nilpotency class 2. In addition, for $p$ a prime, we show that finite $p$-groups $G$, with two distinct irreducible complex character degrees, satisfy the generalized Amit conjecture for words $w_k =[x_1,y_1]...[x_k,y_k]$ with $k$ a natural number; that is, for $g$ a $w_k$-value in $G$ we have $N_{w_k}(g) \ge |G|^{2k-1}$. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

math.GR↗

arXiv2019

Generating pairs of projective special linear groups that fail to lift

The following problem was originally posed by B.H. Neumann and H. Neumann. Suppose that a group $G$ can be generated by $n$ elements and that $H$ is a homomorphic image of $G$. Does there exist, for every generating $n$-tuple $(h_1,\ldots, h_n)$ of $H$, a homomorphism $\vartheta \colon G \to H$ and a generating $n$-tuple $(g_1,\ldots,g_n)$ of $G$ such that $(g_1^\vartheta,\ldots,g_n^\vartheta) = (h_1,\ldots,h_n)$? M.J. Dunwoody gave a negative answer to this question, by means of a carefully engineered construction of an explicit pair of soluble groups. Via a new approach we produce, for $n = 2$, infinitely many pairs of groups $(G,H)$ that are negative examples to the Neumanns' problem. These new examples are easily described: $G$ is a free product of two suitable finite cyclic groups, such as $C_2 \ast C_3$, and $H$ is a suitable finite projective special linear group, such as $\mathrm{PSL}(2,p)$ for a prime $p \ge 5$. A small modification yields the first negative examples $(G,H)$ with $H$ infinite.

math.GR↗

arXiv2019

A pro-p group with infinite normal Hausdorff spectra

Using wreath products, we construct a finitely generated pro-p group G with infinite normal Hausdorff spectrum with respect to the p-power series. More precisely, we show that this normal Hausdorff spectrum contains an infinite interval; this settles a question of Shalev. Furthermore, we prove that the normal Hausdorff spectra of G with respect to other filtration series have a similar shape. In particular, our analysis applies to standard filtration series such as the Frattini series, the lower p-series and the modular dimension subgroup series. Lastly, we pin down the ordinary Hausdorff spectra of G with respect to the standard filtration series. The spectrum of G for the lower p-series displays surprising new features.

math.GR↗

arXiv2019

Hausdorff dimensions in $p$-adic analytic groups

Let $G$ be a finitely generated pro-$p$ group, equipped with the $p$-power series. The associated metric and Hausdorff dimension function give rise to the Hausdorff spectrum, which consists of the Hausdorff dimensions of closed subgroups of $G$. In the case where $G$ is $p$-adic analytic, the Hausdorff dimension function is well understood; in particular, the Hausdorff spectrum consists of finitely many rational numbers closely linked to the analytic dimensions of subgroups of $G$. Conversely, it is a long-standing open question whether the finiteness of the Hausdorff spectrum implies that $G$ is $p$-adic analytic. We prove that the answer is yes, in a strong sense, under the extra condition that $G$ is soluble. Furthermore, we explore the problem and related questions also for other filtration series, such as the lower $p$-series, the Frattini series, the modular dimension subgroup series and quite general filtration series. For instance, we prove, for odd primes $p$, that every countably based pro-$p$ group $G$ with an open subgroup mapping onto 2 copies of the $p$-adic integers admits a filtration series such that the corresponding Hausdorff spectrum contains an infinite real interval.

math.GR↗

arXiv2018

Normal Hausdorff spectra of pro-2 groups

Klopsch and the author have constructed a finitely generated pro-p group G, for p an odd prime, with infinite normal Hausdorff spectrum with respect to the p-power series. They show that the normal Hausdorff spectrum of G contains an infinite interval, which answers a question of Shalev. They indicate in their paper how their results extend to the case p=2. In this note, we provide all the details for the even case.

math.GR↗

arXiv2017

Maximal subgroups and irreducible representations of generalised multi-edge spinal groups

Let $p\ge 3$ be a prime. A generalised multi-edge spinal group is a subgroup of the automorphism group of a regular $p$-adic rooted tree T that is generated by one rooted automorphism and $p$ families of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalises the concepts of multi-edge spinal groups, including the widely studied GGS-groups, and the extended Gupta-Sidki groups that were introduced by Pervova. Extending techniques that were developed in these more special cases, we prove: generalised multi-edge spinal groups that are torsion have no maximal subgroups of infinite index. Furthermore we use tree enveloping algebras, which were introduced by Sidki and Bartholdi, to show that certain generalised multi-edge spinal groups admit faithful infinite dimensional irreducible representations over the prime field $\mathbb{Z}/p\mathbb{Z}$.

math.GR↗

arXiv2016

Maximal subgroups of multi-edge spinal groups

A multi-edge spinal group is a subgroup of the automorphism group of a regular p-adic rooted tree, generated by one rooted automorphism and a finite number of directed automorphisms sharing a common directing path. We prove that torsion multi-edge spinal groups do not have maximal subgroups of infinite index. This generalizes a result of Pervova for GGS-groups.

math.GR↗

arXiv2015

A note on automorphisms of finite $p$-groups

Let $G$ be a finite non-cyclic $p$-group of order at least $p^3$. If $G$ has an abelian maximal subgroup, or if $G$ has an elementary abelian centre with $C_G(Z(Φ(G))) \ne Φ(G)$, then $|G|$ divides $|\text{Aut}(G)|$.

math.GR↗

arXiv2013

Conciseness of coprime commutators in finite groups

Let $G$ be a finite group. We show that the order of the subgroup generated by coprime $γ_k$-commutators (respectively $δ_k$-commutators) is bounded in terms of the size of the set of coprime $γ_k$-commutators (respectively $δ_k$-commutators). This is in parallel with the classical theorem due to Turner-Smith that the words $γ_k$ and $δ_k$ are concise.

math.GR↗

arXiv2011

The Automorphism Group of p-Central p-Groups

A p-group G is p-central if the central quotient has exponent p. We prove that for a subset of finite p-central p-groups, the order of the group G divides the order of Aut(G).

math.GR↗

arXiv2011

Groups of p-deficiency one

The main result of [4] is that all finitely presented groups of p-deficiency greater than one are p-large. Here we prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto . This implies that these groups do not have Kazhdan's property (T). Additionally, we prove that the main result of [4] implies a result of Lackenby [8].

math.GR↗