arXiv subjects
Publications and source records attributed to Diego Rattaggi.
The computation of Dalzell integrals $\int_0^1 \frac{x^m (1-x)^n}{1+x^2} \, dx > 0$ gives new error estimates for the partial sums of the Gregory-Leibniz series $1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} \pm \ldots$ and for the alternating harmonic series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \pm \ldots$
Let $x$, $y$ be two integral quaternions of norm $p$ and $l$, respectively, where $p$, $l$ are distinct odd prime numbers. We investigate the structure of $ $, the multiplicative group generated by $x$ and $y$. Under a certain condition which excludes $ $ from being free or abelian, we show for example that $ $, its center, commutator subgroup and abelianization are finitely presented infinite groups. We give many examples where our condition is satisfied and compute as an illustration a finite presentation of the group $<1+j+k, 1+2j>$ having these two generators and seven relations. In a second part, we study the basic question whether there exist commuting quaternions $x$ and $y$ for fixed $p$, $l$, using results on prime numbers of the form $r^2 + m s^2$ and a simple invariant for commutativity.
Let $G_1 \times G_2$ be a subgroup of $\mathrm{SO}_3(\mathbb{R})$ such that the two factors $G_1$ and $G_2$ are non-trivial groups. We show that if $G_1 \times G_2$ is not abelian, then one factor is the (abelian) group of order 2, and the other factor is non-abelian and contains an element of order 2. There exist finite and infinite such non-abelian subgroups.
In the first part, we prove that the number of $(1,β)$-BM relations is $3 \cdot 5 \cdot \ldots \cdot (2β+ 1)$, which was conjectured by Kimberley. In the second part, we construct two isomorphisms between certain $(2,2)$-BM groups. This completes the classification of $(2,2)$-BM groups initiated by Kimberley and Robertson.
We extend a conjecture of Kimberley-Robertson on the abelianizations of certain square complex groups.
An anti-torus is a subgroup $ $ in the fundamental group of a compact non-positively curved space $X$, acting in a specific way on the universal covering space $\tilde{X}$ such that $a$ and $b$ do not have any commuting non-trivial powers. We construct and investigate anti-tori in a class of commutative transitive fundamental groups of finite square complexes, in particular for the groups $Γ_{p,l}$ originally studied by Mozes [15]. It turns out that anti-tori in $Γ_{p,l}$ directly correspond to non-commuting pairs of Hamilton quaternions. Moreover, free anti-tori in $Γ_{p,l}$ are related to free groups generated by two integer quaternions, and also to free subgroups of $\mathrm{SO}_3(\mathbb{Q})$. As an application, we prove that the multiplicative group generated by the two quaternions $1+2i$ and $1+4k$ is not free.
We give an example of a finitely presented simple group containing a finitely generated subgroup which is not finitely presented.
We construct three groups $Λ_1$, $Λ_2$, $Λ_3$, which can all be decomposed as amalgamated products $F_9 \ast_{F_{81}} F_{9}$ and have very few normal subgroups of finite or infinite index. Concretely, $Λ_1$ is a simple group, $Λ_2$ is not simple but has no non-trivial normal subgroup of infinite index, and $Λ_3$ is not simple but has no proper subgroup of finite index.
A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group $Γ$ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of $Γ$ is studied in some detail.
We construct a finitely presented torsion-free simple group $Σ_0$, acting cocompactly on a product of two regular trees. An infinite family of such groups has been introduced by Burger-Mozes ([2,4]). We refine their methods and get $Σ_0$ as an index 4 subgroup of a group $Σ< \mathrm{Aut}(\mathcal{T}_{12}) \times \mathrm{Aut}(\mathcal{T}_{8})$ presented by 10 generators and 24 short relations. For comparison, the smallest virtually simple group of [4, Theorem 6.4] needs more than 18000 relations, and the smallest simple group constructed in [4, Section 6.5] needs even more than 360000 relations in any finite presentation.