On mennicke groups of deficiency zero I

The Mennicke group M(m,n,r)=〈x,y,z,|xy=xm, yz=yn, zx=zr〉 is one of the few known 3-generator groups of deficiency zero. Several cases of M(m,n,r) are studied

Groups of Order 2048 with Three Generators and Three Relations

It is shown that there are exactly seventy-eight 3-generator 2- groups of order 2^11 with trivial Schur multiplier. We then give 3-generator, 3-relation presentations for forty-eight of them proving that these groups have deficiency zero

Maximal unramified 3-extensions of imaginary quadratic fields and SL_2(Z_3)

The structure of the Galois group of the maximal unramified p-extension of an imaginary quadratic field is restricted in various ways. In this paper we construct a family of finite 3-groups satisfying these restrictions. We prove several results about this family and characterize them as finite extensions of certain quotients of a Sylow pro-3 subgroup of SL_2(Z_3). We verify that the first group in the family does indeed arise as such a Galois group and provide a small amount of evidence that this may hold for the other members. If this were the case then it would imply that there is no upper bound on the possible lengths of a finite p-class tower.Comment: 7 pages. No figures. LaTe

Structure of the Macdonald groups in one parameter

Consider the Macdonald groups $G(\alpha)=\langle A,B\,|\, A^{[A,B]}=A^\alpha,\, B^{[B,A]}=B^\alpha\rangle$, $\alpha\in\Z$. We fill a gap in Macdonald's proof that $G(\alpha)$ is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of $G(\alpha)$

Minimal Euler Characteristics for Even-Dimensional Manifolds with Finite Fundamental Group

We consider the Euler characteristics $\chi(M)$ of closed orientable topological $2n$-manifolds with $(n-1)$-connected universal cover and a given fundamental group $G$ of type $F_n$. We define $q_{2n}(G)$, a generalized version of the Hausmann-Weinberger invariant for 4-manifolds, as the minimal value of $(-1)^n\chi (M)$. For all $n\geq 2$, we establish a strengthened and extended version of their estimates, in terms of explicit cohomological invariants of $G$. As an application we obtain new restrictions for non-abelian finite groups arising as fundamental groups of rational homology 4-spheres.Comment: 24 pages (v3): Improvements to exposition; new statement and proof for Theorem 3.8; Details for Example 5.9 added in Appendix B; Acknowledgement to Mike Newman and Ozgun Unlu for information about deficiency zero example

Nilpotent groups with balanced presentations. II

If $G$ is a nilpotent group with a balanced presentation and $G\not\cong\mathbb{Z}^3$ then $\beta_1(G;\mathbb{Q})\leq2$ \cite{Hi22}. We show that if such a group $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}^2$ then $G$ is torsion-free and has Hirsch length $h(G)\leq4$. On the other hand, if $\beta_1(G;\mathbb{Q})=1$ and $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}$ then $G\cong\mathbb{Z}/m\mathbb{Z}\rtimes_n\mathbb{Z}$, for some $m,n\not=0$ such that $m$ divides a power of $n-1$.Comment: v3: completely recast, following blunder in use of Wang sequence. v4 New title, reflecting a shift in emphasis; substantially rewritten and enlarged. v5: further reorganisation, sharper final result, final section deleted. v6: reorganised to emphasise algebra over topology (new abstract), v7 final section deleted for use elsewhere, minor changes to Theorem 1

Finite groups defined by presentations in which each defining relator involves exactly two generators

We consider two classes of groups, denoted JΓ and MΓ, defined by presentations in which each defining relator involves exactly two generators, and so are examples of simple Pride groups. (For MΓ the relators are Baumslag-Solitar relators.) These presentations are, in turn, defined in terms of a non-trivial, simple directed graph Γ whose arcs are labelled by integers. When Γ is a directed triangle the groups JΓ,MΓ coincide with groups considered by Johnson and by Mennicke, respectively. When the arc labels are all equal the groups form families of so-called digraph groups. We show that if Γ is a non-trivial, strongly connected tournament then the groups JΓ are finite, metabelian, of rank equal to the order of Γ and we show that the groups MΓ are finite and, subject to a condition on the arc labels, are of rank equal to the order of Γ