Anti-tori in Square Complex Groups

An anti-torus is a subgroup 〈a,b 〉 in the fundamental group of a compact non-positively curved space X, acting in a specific way on the universal covering space X such that a and b do not have any commuting nontrivial powers. We construct and investigate anti-tori in a class of commutative transitive fundamental groups of finite square complexes, in particular for the groups Γp,l originally studied by Mozes [Israel J. Math. 90(1-3) (1995), 253-294]. It turns out that anti-tori in Γp,l directly correspond to non commuting pairs of Hamilton quaternions. Moreover, free anti-tori in Γp,l are related to free groups generated by two integer quaternions, and also to free subgroups of $$SO_3(\mathbb{Q})$$ . As an application, we prove that the multiplicative group generated by the two quaternions 1+2i and 1+4k is not fre

On infinite groups generated by two quaternions

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 $$, its center, commutator subgroup and abelianization are finitely presented infinite groups. We give many examples where our condition is satisfied and compute as an illustration a finite presentation of the group$$ 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.Comment: 31 pages. Completely revised version, several new results and simplified proof

On the commutator of unit quaternions and the numbers 12 and 24

The quaternions are non-commutative. The deviation from commutativity is encapsulated in the commutator of unit quaternions. It is known that the k-th power of the commutator is null-homotopic if and only if k is divisible by 12. The main purpose of this paper is to construct a concrete null-homotopy of the 12-th power of the commutator. Subsequently, we construct free S^3-actions on S^7 x S^3 whose quotients are exotic 7-sphere and give a geometric explanation for the order of the stable homotopy groups \pi_{n+3} (S^n). Intermediate results of perhaps independent interest are a construction of the octonions emphasizing the inclusion SU(3) \subset G_2, a detailed study of Duran's geodesic boundary map construction, and explicit formulas for the characteristic maps of the bundles G_2 \to S^6 and Spin(7) \to S^7.Comment: 27 pages, 2 references added, minor change

A Cancellation Theorem for Modules over Integral Group Rings

A long standing problem, which has it roots in low-dimensional homotopy theory, is to classify all finite groups $G$ for which $\mathbb{Z}[G]$ has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that $\mathbb{Z}[G]$ has SFC provided at most one copy of the quaternions $\mathbb{H}$ occurs in the Wedderburn decomposition of $\mathbb{R}[G]$. This generalises the Eichler condition in the case of integral group rings

Explicit closed-form parametrization of SU(3) and SU(4) in terms of complex quaternions and elementary functions

Remarkably simple closed-form expressions for the elements of the groups SU(n), SL(n,R), and SL(n,C) with n=2, 3, and 4 are obtained using linear functions of biquaternions instead of n x n matrices. These representations do not directly generalize to SU(n>4). However, the quaternion methods used are sufficiently general to find applications in quantum chromodynamics and other problems which necessitate complicated 3 x 3 or 4 x 4 matrix calculations.Comment: Submitted to Journal of Mathematical Physics, 17 pages, 1 table

Quaternions, polarizations and class numbers

We study abelian varieties $A$ with multiplication by a totally indefinite quaternion algebra over a totally real number field and give a criterion for the existence of principal polarizations on them in pure arithmetic terms. Moreover, we give an expression for the number $\pi_0(A)$ of isomorphism classes of principal polarizations on $A$ in terms of relative class numbers of CM fields by means of Eichler's theory of optimal embeddings. As a consequence, we exhibit simple abelian varieties of any even dimension admitting arbitrarily many non-isomorphic principal polarizations. On the other hand, we prove that $\pi_0(A)$ is uniformly bounded for simple abelian varieties of odd square-free dimension.Comment: To appear in Crell

The Quaternionic Affine Group and Related Continuous Wavelet Transforms on Complex and Quaternionic Hilbert Spaces

By analogy with the real and complex affine groups, whose unitary irreducible representations are used to define the one and two-dimensional continuous wavelet transforms, we study here the quaternionic affine group and construct its unitary irreducible representations. These representations are constructed both on a complex and a quaternionic Hilbert space. As in the real and complex cases, the representations for the quaternionic group also turn out to be square-integrable. Using these representations we constrct quaternionic wavelets and continuous wavelet transforms on both the complex and quaternionic Hilbert spaces.Comment: 15 page

Integer-quaternion formulation of Lambek's representation of fundamental particles and their interactions

Lambek's unified classification of the elementary interaction-quanta of the ``Standard model'' is formulated in terms of the 24 units of the integer-quaternion ring, i.e., the tetrahedral group Q_{24}. An extension of Lambek's scheme to the octahedral group Q_{48} may enable to take all three generations of leptons and quarks into account, as well as to provide a quantitative explanation for flavor-mixing.Comment: 10 pages, 3 tables. Error in the abstract correcte

An Invitation to Noncommutative Algebra

This is a brief introduction to the world of Noncommutative Algebra aimed at advanced undergraduate and beginning graduate students.Comment: v2: Minor edits. To appear in the EDGE program volume for the AWM Springer series. 27 pages, including 14 figures, references, and a note on connection to EDGE progra

Geometric enumeration problems for lattices and embedded Z\mathbb{Z}-modules

In this review, we count and classify certain sublattices of a given lattice, as motivated by crystallography. We use methods from algebra and algebraic number theory to find and enumerate the sublattices according to their index. In addition, we use tools from analytic number theory to determine the asymptotic behaviour of the corresponding counting functions. Our main focus lies on similar sublattices and coincidence site lattices, the latter playing an important role in crystallography. As many results are algebraic in nature, we also generalise them to $\mathbb{Z}$-modules embedded in $\mathbb{R}^d$.Comment: 92 pages, 2 figures; review articl