In 1930, Wilhelm Magnus introduced the so-called Freiheitssatz: Let $F$ be a free group with basis $\mathcal{X}$ and let $r$ be a cyclically reduced element of $F$ which contains a basis element $x \in \mathcal{X}$, then every non-trivial element of the normal closure of $r$ in $F$ contains the basis element $x$. Equivalently, the subgroup freely generated by $\mathcal{X} \backslash \{x\}$ embeds canonically into the quotient group $F / \langle \! \langle r \rangle \! \rangle_{F}$. In this article, we want to introduce a Freiheitssatz for amalgamated products $G=A \ast_{U} B$ of free groups $A$ and $B$, where $U$ is a maximal cyclic subgroup in $A$ and $B$: If an element $r$ of $G$ is neither conjugate to an element of $A$ nor $B$, then the factors $A$, $B$ embed canonically into $G / \langle \! \langle r \rangle \! \rangle_{G}$.
We discuss the linearity and discreteness of amalgamated products of linear word-hyperbolic groups. In particular, we prove that the double of an Anosov group along a maximal cyclic subgroup is always linear, and we construct examples of such groups which do not admit any discrete and faithful representation in rank 1. We also build new examples of non-linear word-hyperbolic groups, elaborating on a previous work of Canary--Stover--Tsouvalas.
Garside groups are a natural lattice-theoretic generalisation of the braid groups and spherical type Artin--Tits groups. Here we show that the class of Garside groups is closed under some free products with cyclic amalgamated subgroups. We deduce that every tree product of infinite cyclic groups is a Garside group. Moreover, we study those cyclic HNN extensions of Garside groups that are Garside groups as well. Using a theorem of Pietrowski, we conclude this paper by stating that a non-cyclic one-relator group is Garside if and only if its centre is nontrivial.
We construct the type-preserving panel-regular lattices in buildings of type M, for M with entries two, three or infinity, which have cyclic stabilizers of spherical 2-residues. We obtain these lattices as fundamental groups of their associated quotient buildings, which are constructed by amalgamating quotients of generalized digons and triangles by actions of Singer cyclic groups. Each quotient construction has an associated gluing matrix which encodes how the quotient generalized polygons are amalgamated. We show that these gluing matrices also encode presentations of the panel-regular latti…
A finitely generated group G is termed parafree if it is residually nilpotent and it has the same isomorphism types of nilpotent quotients as some free group. The two main results of this MSc. Thesis characterise the parafreeness of two constructions, namely amalgamated products with cyclic amalgams and cyclic HNN extensions.
We classify all subgroups of $SO(3)$ that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of $π$. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple consequences of standard facts about rotations by $π$ and $π/2$. Embedded in the subgroups are explicit free groups on 2 generators, as used in the Banach-Tarski paradox.
We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to partially commutative groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element $r$ of a partially commutative group $\mathbb{G}$, certain Magnus subgroups embed in the quotient $G=\mathbb{G}/N(r)$; that if $r=s^n$ has root $s$ in $\mathbb{G}$ then the order of $s$ in $G$ is $n$, and under slightly stronger conditions that the word problem of $G$ is decidable. We also give conditions under which the question of which Magnus subgroups of $\…
Groups with Context-Free Co-Word Problem and Embeddings into Thompson's Group $V$
设$G$是一个群生成的群，$Σ$是一个生成$G$为幺半群的有限子集。\emph{word problem of $G$ with respect to $Σ$}由自由幺半群$Σ^{\ast}$中所有与$G$中的恒等式相等的字组成。\emph{co-word problem of $G$ with respect to $Σ$}是字问题在$Σ^{\ast}$中的补充。我们说一个群$G$是\emph{co$\mathcal{CF}$}，如果它关于某个（等价地，任何）有限生成集$Σ$的共字问题是上下文无关语言。我们对每个有限群G描述了一个广义Thompson群$V_{(G, θ)}$和同态$θ$：$G \rightarrow G$ .我们的群体是使用Witzel和Zaremsky介绍的克隆系统构建的。通过构造一个下推自动机，证明了对于任意同态$θ$和有限群G，$V_{(G, θ)}$是coPDMATHABMV，并证明了$V_{(G, θ)}$的co-字问题是我们的自动机所接受的语言的循环移位.一个版本的猜想，由于Lehnert说，一个组有上下文无关的共同字问题正是如果它是一个生成子群的V.组$V_{(G,θ)}$，其中$θ$不是单位同态似乎没有明显的嵌入到V，因此可能被视为反例的猜想。由Bleak和Salazar-Diaz引入的$V$的示范子群可以用来构造某些环积和合并自由积到$V$中的…
We study $K$-theoretical aspects of the braid groups $B\_n(\mathbb{S}^{2})$ on $n$ strings of the $2$-sphere, which by results of the second two authors, are known to satisfy the Farrell-Jones fibred isomorphism conjecture~\cite{JM}. In light of this, in order to determine the algebraic $K$-theory of the group ring $\mathbb{Z}[B\_n(\mathbb{S}^{2})]$, one should first compute that of its virtually cyclic subgroups, which were classified by D.~L.~Gon{\c c}alves and the first author. We calculate the Whitehead and $K\_{-1}$-groups of the group rings of the finite subgroups (dicyclic and binary po…
For an integer $M\geq 2$ and a finite group $G$, an element $α\in G$ is called an $M$-th power if it satisfies $A^M=α$ for some $A\in G$. In this article, we will deal with the case when $G$ is finite symplectic or orthogonal group over a field of order $q$. We introduce the notion of $M^*$-power SRIM polynomials. This, amalgamated with the concept of $M$-power polynomial, we provide the complete classification of the conjugacy classes of regular semisimple, semisimple, cyclic and regular elements in $G$, which are $M$-th powers, when $(M,q)=1$. The approach here is of generating functions, as…
Parity patterns associated with lifts of Hecke groups
令$q$为奇素数，$m$为正整数，且令$\Ga_m(q)$为由两个元素$x$和$y$服从关系$x^{2m}=y^{qm}=1$和$x^2=y^q$生成的群;即，$\Ga_m(q)$为两个分别为$2m$和$qm$的循环群的自由积，沿着它们的$m$的子群合并。我们的主要结果确定了[T. W. Müller，Adv. in Math.153（2000），118-154]，并且在当$\gcd(m,| H|)=1$ .这个计算依赖于Hecke群$\mathfrak H(q)=C_2*C_q$中三个计数问题的解：（i）确定$\mathfrak H(q)$的子群数的奇偶性;（ii）确定与$C_2$和$C_\infty$的拷贝的自由积同构的$\mathfrak H(q)$的指数$n$子群数的奇偶性;（iii）确定$\mathfrak H(q)$中与$C_q$拷贝的自由产物同构的指数$n$子群的数目的奇偶性。第一个问题在[T. W. Müller，in：{ \it Groups：Topological，Combinatorial and Arithmetic Aspects}（T. W. Müller编），LMS Lecture Notes Series 311，Cambridge University Press，Cambridge，2004，pp. 327-374]。本文的大部分内容涉及问题（i…
