Groups of prime-power order with a small second derived quotient

Zariski density and finite quotients of mapping class groups

Our main result is that the image of the quantum representation of a central extension of the mapping class group of the genus $g\geq 3$ closed orientable surface at a prime $p\geq 5$ is a Zariski dense discrete subgroup of some higher rank algebraic semi-simple Lie group $\mathbb G_p$ defined over $\Q$. As an application we find that, for any prime $p\geq 5$ a central extension of the genus $g$ mapping class group surjects onto the finite groups $\mathbb G_p(\Z/q\Z)$, for all but finitely many primes $q$. This method provides infinitely many finite quotients of a given mapping class group out…

Small derived quotients in finite p-groups

More than 70 years ago, P. Hall showed that if $G$ is a finite $p$-group such that a term $\der G{d+1}$ of the derived series is non-trivial, then the order of the quotient $\der Gd/\der G{d+1}$ is at least $p^{2^d+1}$. Recently Mann proved that, in a finite $p$-group, Hall's lower bound can be taken for at most two distinct $d$. We improve this result and show that if $p$ is odd, then it can only be taken for two distinct $d$ in a group with order $p^6$.

On groups of finite upper rank

The `upper rank' of a group is the supremum of the (Prüfer) ranks of its finite quotients, and for a prime $p$, the `upper $p$-rank' is the supremum of the sectional $p$-ranks of those quotients. The former is finite if and only if the latter are finitely bounded as $p$ ranges over all primes (a deep fact). Here we discuss the question: if the upper $p$-ranks of a finitely generated group $G$ are all finite, are they necessarily bounded? The case where $G$ is a soluble group is still an open problem.

Multi-EGS-groups：同余商的指数

Multi-EGS-groups: exponent of congruence quotients

本文研究了作用于p进根树的multi-EGS-group，推导出计算所有同余商指数的公式，并证明该公式仅依赖于素数p、层数n及群的周期性。

Cohomology of Toroidal Orbifold Quotients

Let $ϕ:\Z/p\to GL_{n}(\Z)$ denote an integral representation of the cyclic group of prime order $p$. This induces a $\Z/p$-action on the torus $X=\R^{n}/\Z^{n}$. The goal of this paper is to explicitly compute the cohomology groups $H^{*}(X/\Z/p;\Z)$ for any such representation. As a consequence we obtain an explicit calculation of the integral cohomology of the classifying space associated to the family of finite subgroups for any crystallographic group $Γ=\Z^n\rtimes\Z/p$ with prime holonomy.

具有素数幂指数可解子群的群

Groups with a solvable subgroup of prime-power index

本文研究了包含有限素数幂指数可解子群的群的性质，证明了此类群的商群在某些条件下会渐近趋于小。

Pronilpotent quotients associated with primitive substitutions

We describe the pronilpotent quotients of a class of projective profinite groups, that we call $ω$-presented groups, defined using a special type of presentations. The pronilpotent quotients of an $ω$-presented group are completely determined by a single polynomial, closely related with the characteristic polynomial of a matrix. We deduce that $ω$-presented groups are either perfect or admit the $p$-adic integers as quotients for cofinitely many primes. We also find necessary conditions for absolute and relative freeness of $ω$-presented groups. Our main motivation comes from semigroup theory:…

On exceptional groups of order p^5

A finite group G is exceptional if it has a quotient Q whose minimal faithful permutation degree is greater than that of G. We say that Q is a distinguished quotient. The smallest examples of exceptional p-groups have order p^5. For an odd prime p, we classify all pairs (G,Q) where G has order p^5 and Q is a distinguished quotient. (The case p=2 has already been treated by Easdown and Praeger.) We establish the striking asymptotic result that as p increases, the proportion of groups of order p^5 with at least one exceptional quotient tends to 1/2.

A remark on groups without finite quotients

We notice that the class of nontrivial groups without proper subgroups of finite index is not elementary, because some groups in this class, such as $\mathbb Q*\mathbb Q$, have ultrapowers that map homomorphically onto $\mathbb Z/p\mathbb Z$ for every prime $p$. Also, some ultrapowers of certain simple groups map homomorphically onto $\mathbb Z/2\mathbb Z$.

Distinguishing some genus one knots using finite quotients

We give a criterion for distinguishing a prime knot $K$ in $S^3$ from every other knot in $S^3$ using the finite quotients of $π_1(S^3\setminus K)$. Using recent work of Baldwin-Sivek, we apply this criterion to the hyperbolic knots $5_2$, $15n_{43522}$, and the three-strand pretzel knots $P(-3,3,2n+1)$ for every integer $n$.

Some behaviors of FSZ groups under central products, central quotients, and regular wreath products

We show that any group $G$ with a non-$FSZ_m$ quotient by a central cyclic subgroup also provides a non-$FSZ_m$ group of order $m|G|$ obtained as a central product of $G$ with a cyclic group. We then construct, for every prime $p>3$ and $j\in\mathbb{N}$, an $FSZ_{p^j}$ group $F$ such that there is a central cyclic subgroup $A$ with $F/A$ not $FSZ_{p^j}$. We apply these results to regular wreath products to construct an $FSZ$ $p$-group which is not $FSZ^+$ for any prime $p>3$. These give the first known examples of $FSZ$ groups that are not $FSZ^+$. We are also able to prove a few partial resul…

Finite quotients of Galois pro-$p$ groups and rigid fields

For a prime number $p$, we show that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-$p$ group $G$ coincide, then $G$ has a very simple structure, i.e., $G$ is a $p$-adic analytic pro-$p$ group. This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions $F^{(3)}/F$ and $F^{\{3\}}/F$ of a field $F$ -- with $F$ containing a primitive $p$-th root of unity -- coincide, then $F$ is $p$-rigid. The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-$p$ groups.

On finite derived quotients of 3-manifold groups

This paper studies the set of finite groups appearing as $π_1(M)/π_1(M)^{(n)}$, where $M$ is a closed, orientable 3-manifold and $π_1(M)^{(n)}$ denotes the $n$-th term of the derived series of $π_1(M)$. Our main result is that if $M$ is a closed, orientable 3-manifold, $n\ge 2$, and $G\cong π_1(M)/π_1(M)^{(n)}$ is finite, then the cup product pairing $H^2(G)\otimes H^2(G)\to H^4(G)$ has cyclic image $C$, and the pairing $H^2(G)\otimes H^2(G)\stackrel{\smile}{\longrightarrow} C$ is isomorphic to the linking pairing $H_1(M)_{\textrm{Tors}}\otimes H_1(M)_{\textrm{Tors}}\to \mathbb{Q}/\mathbb{Z}$.

The Cohomology of canonical quotients of free groups and Lyndon words

For a prime number $p$ and a free profinite group $S$, let $S^{(n,p)}$ be the $n$th term of its lower $p$-central filtration, and $S^{[n,p]}$ the corresponding quotient. Using tools from the combinatorics of words, we construct a canonical basis of the cohomology group $H^2(S^{[n,p]},\mathbb{Z}/p)$, which we call the Lyndon basis, and use it to obtain structural results on this group. We show a duality between the Lyndon basis and canonical generators of $S^{(n,p)}/S^{(n+1,p)}$. We prove that the cohomology group satisfies shuffle relations, which for small values of $n$ fully describe it.

Quotients of absolute Galois groups which determine the entire Galois cohomology

For prime power $q=p^d$ and a field $F$ containing a root of unity of order $q$ we show that the Galois cohomology ring $H^*(G_F,\dbZ/q)$ is determined by a quotient $G_F^{[3]}$ of the absolute Galois group $G_F$ related to its descending $q$-central sequence. Conversely, we show that $G_F^{[3]}$ is determined by the lower cohomology of $G_F$. This is used to give new examples of pro-$p$ groups which do not occur as absolute Galois groups of fields.