arXiv subjects

Emilio Pierro

Publications and source records attributed to Emilio Pierro.

8 recordsLinked to original sources

arXiv2021

Hurwitz Generation in Groups of Types $F_4$, $E_6$, $^2E_6$, $E_7$ and $E_8$

A Hurwitz generating triple for a group $G$ is an ordered triple of elements $(x,y,z) \in G^3$ where $x^2=y^3=z^7=xyz=1$ and $\langle x,y,z \rangle = G$. For the finite quasisimple exceptional groups of types $F_4$, $E_6$, $^2E_6$, $E_7$ and $E_8$, we provide restrictions on which conjugacy classes $x$, $y$ and $z$ can belong to if $(x,y,z)$ is a Hurwitz generating triple. We prove that there exist Hurwitz generating triples for $F_4(3)$, $F_4(5)$, $F_4(7)$, $F_4(8)$, $E_6(3)$ and $E_7(2)$, and that there are no such triples for $F_4(2^{3n-2})$, $F_4(2^{3n-1})$, $E_6(7^{3n-2})$, $E_6(7^{3n-1})$, $SE_6(7^n)$ or $^2E_6(7^n)$ when $n \geq 1$.

math.GR↗

arXiv2021

The order complex of $PGL_2(p^{2^n})$ is contractible when $p$ is odd

Given a group $G$, its lattice of subgroups $\mathcal{L}(G)$ can be viewed as a simplicial complex in a natural way. The inclusion of $1_G, G \in \mathcal{L}(G)$ implies that $\mathcal{L}(G)$ is contractible, and so we study the topology of the order complex $\widehat{\mathcal{L}(G)} := \mathcal{L}(G) \setminus \{1_G,G\}$. In this short note we consider the homotopy type of $\widehat{\mathcal{L}(G)}$ where $G \cong PGL_2(p^{2^n})$, $p \geq 3$, $n \geq 1$ and show that $\widehat{\mathcal{L}(G)}$ is contractible. This is consistent with a conjecture of Shareshian on the homotopy type of order complexes of finite groups.

math.CO↗

arXiv2019

On the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$

We show that the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$ is $\mathrm{PSL}_n(\mathbb{Z}/2\mathbb{Z}) = \mathrm{L}_n(2)$, thus confirming a conjecture of Mecchia--Zimmermann. In the course of the proof we give an exponential (in $n$) lower bound for the cardinality of a set on which $\mathrm{SAut}(F_n)$, the unique index $2$ subgroup of $\mathrm{Aut}(F_n)$, can act non-trivially. We also offer new results on the representation theory of $\mathrm{SAut(F_n)}$ in small dimensions over small, positive characteristics, and on rigidity of maps from $\mathrm{SAut}(F_n)$ to finite groups of Lie type and algebraic groups in characteristic $2$.

math.GR↗

arXiv2019

On the smallest non-trivial quotients of mapping class groups

We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus at least 3 without punctures is $\mathrm{Sp}_{2g}(2)$, thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz's results on $\mathbb{C}$-linear representations of mapping class groups to projective representations over any field.

math.GR↗

arXiv2018

The Hurwitz Subgroups of $E_6(2)$

We prove that the exceptional group $E_6(2)$ is not a Hurwitz group. In the course of proving this, we complete the classification up to conjugacy of all Hurwitz subgroups of $E_6(2)$, in particular, those isomorphic to $L_2(8)$ and $L_3(2)$.

math.GR↗

arXiv2017

Doubly Hurwitz Beauville groups

If $\mathcal S$ is a Beauville surface $({\mathcal C}_1\times{\mathcal C}_2)/G$, then the Hurwitz bound implies that $|G|\le 1764\,χ({\mathcal S})$, with equality if and only if the Beauville group $G$ acts as a Hurwitz group on both curves ${\mathcal C}_i$. Equivalently, $G$ has two generating triples of type $(2,3,7)$, such that no generator in one triple is conjugate to a power of a generator in the other. We show that this property is satisfied by alternating groups $A_n$, their double covers $2.A_n$, and special linear groups $SL_n(q)$ if $n$ is sufficiently large, but by no sporadic simple groups or simple groups $L_n(q)$ ($n\le 7$), ${}^2G_2(3^e)$, ${}^2F_4(2^e)$, ${}^2F_4(2)'$, $G_2(q)$ or ${}^3D_4(q)$ of small Lie rank.

math.GR↗

arXiv2015

The Möbius function of the small Ree groups

The Möbius function for a group, $G$, was introduced in 1936 by Hall in order to count ordered generating sets of $G$. In this paper we determine the Möbius function of the simple small Ree groups, $R(q)={}^2G_2(q)$ where $q=3^{2m+1}$ for $m>0$, using their 2-transitive permutation representation of degree $q^3+1$ and describe their maximal subgroups in terms of this representation. We then use this to determine $\vert$Epi$(Γ,G)\vert$ for various $Γ$, such as $F_2$ or the modular group $PSL_2(\mathbb{Z})$, with applications to Grothendieck's theory of dessins d'enfants as well as probabilistic generation of the small Ree groups.

math.GR↗

arXiv2014

New Examples of Mixed Beauville Groups

We generalise a construction of mixed Beauville groups first given by Bauer, Catanese and Grunewald. We go on to give several examples of infinite families of characteristically simple groups that satisfy the hypotheses of our theorem and thus provide a wealth of new examples of mixed Beauville groups.

math.GR↗