Bianchi Groups Research Articles

On multiplicities in length spectra of arithmetic hyperbolic three-orbifolds

Asymptotic laws for mean multiplicities of lengths of closed geodesics in arithmetic hyperbolic three-orbifolds are derived. The sharpest results are obtained for non-compact orbifolds associated with the Bianchi groups and some congruence subgroups. Similar results hold for cocompact arithmetic quaternion groups, if a conjecture on the number of gaps in their length spectra is true. The results related to the groups above give asymptotic lower bounds for the mean multiplicities in length spectra of arbitrary arithmetic hyperbolic three-orbifolds. The investigation of these multiplicities is motivated by their sensitive effect on the eigenvalue spectrum of the Laplace - Beltrami operator on a hyperbolic orbifold, which may be interpreted as the Hamiltonian of a three-dimensional quantum system being strongly chaotic in the classical limit. PACS numbers: 0210, 0240, 0320, 0545

Non-normal, standard subgroups of the Bianchi groups

Let $S$ be a subgroup of $SL_n(K)$, where $K$ is a Dedekind ring, and let $\mathbf {q}$ be the $K$-ideal generated by $x_{ij},x_{ii}-x_{jj}$ $(i\ne j)$, where $(x_{ij})\in S$. The subgroup $S$ is called standard iff $S$ contains the normal subgroup of $SL_n(K)$ generated by the $\mathbf {q}$-elementary matrices. It is known that, when $n\ge 3$, $S$ is standard iff $S$ is normal in $SL_n(K)$. It is also known that every standard subgroup of $SL_2(K)$ is normal in $SL_2(K)$ when $K$ is an arithmetic Dedekind domain with infinitely many units. The ring of integers of an imaginary quadratic number field, $\mathcal {O}$, is one example (of three) of such an arithmetic domain with finitely many units. In this paper it is proved that every Bianchi group $SL_2(\mathcal {O})$ has uncountably many non-normal, standard subgroups. This result is already known for related groups like $SL_2(\mathbb {Z})$.

Fuchsian Subgroups of Bianchi Groups

Fuchsian Subgroups of Bianchi Groups

Diophantine Approximation On Bianchi Groups

A modification of the Ford geometric approach to the problem of approximation of complex numbers by elements of an imaginary quadratic number field is developed. An upper bound for the Hurwitz constant for the field is obtained in terms of the geometry of the isometric fundamental domain for the corresponding Bianchi group. The Hurwitz constants for the fields of discriminant −20 and −24 are found.

Free quotients of subgroups of the Bianchi groups whose kernels contain many elementary matrices

AbstractLet d be a square-free positive integer and let be the ring of integers of the imaginary quadratic number field ℚ(√ − d) The Bianchi groups are the groups SL2() (or PSL2(). Let m be the order of index m in . In this paper we prove that for each d there exist infinitely many m for which SL2(m)/NE2(m) has a free, non-cyclic quotient, where NE2(m) is the normal subgroup of SL2(m) generated by the elementary matrices. When d is not a prime congruent to 3 (mod 4) this result is true for all but finitely many m. The proofs are based on the fundamental paper of Zimmert and its generalization due to Grunewald and Schwermer.The results are used to extend earlier work of Lubotzky on non-congruence subgroups of SL2(), which involves the concept of the ‘non-congruence crack’. In addition the results highlight a number of low-dimensional anomalies. For example, it is known that [SLn(m), SLnm)] = En(m), when n ≥ 3, where [SLn(m), SLn(m)] is the commutator subgroup of SL(m) and En(m) is the subgroup of SLn(m) generated by the elementary matrices. Our results show that this is not always true when n = 2.

Normal Subgroups of Level Zero of the Bianchi Groups

Normal Subgroups of Level Zero of the Bianchi Groups

Reflections in extended Bianchi groups

Letdbe a positive square-free number. LetOdbe the ring of integers in Q(√ −d). The groupsPSL(2, Od) are called collectively the Bianchi groups. The extended Bianchi groupBdis the maximal discrete extension ofPSL(2,Od) inPSL(2, C). The groupBdacts by linear fractional transformations on the complex plane C. LetRBdbeBdwith the generator θ,, adjoined. (RBdwill be also called an extended Bianchi group (cf. [18])). A group is said to be reflective if it contains a Coxeter subgroup (i.e. a subgroup generated by reflections) of finite index. The groupsRBdand their subgroups have been investigated in [3, 6, 13, 15, 16, 17, 18, 20, 21]. In 1892 Bianchi[3] proved thatPGL(2,Od)⋊{θ} is reflective ifd≤ 19,d╪ 14 or 17. Vinberg [18] proved that if the groupRBdis reflective, then the orders of all elements of the ideal-class group of the field Q(√ −d) should divide 4. Shaiheev [16] proved that there are only finitely many reflective extended Bianchi groups and found all of them ford≤ 30. Similar results are obtained for groupsPGL(2,Od) ⋊ {θ} which, as Shvartsman [17] showed, are reflective only whend= 1, 2, 5, 6, 10, 13, 21, providedd= 1 or 2 (mod 4).

Subgroups of Bianchi Groups and Arithmetic Quotients of Hyperbolic 3-Space

Subgroups of Bianchi Groups and Arithmetic Quotients of Hyperbolic 3-Space

Subgroups of Bianchi groups and arithmetic quotients of hyperbolic 3-space

Let O \mathcal {O} be the ring of integers in an imaginary quadratic number-field. The group PSL 2 ( O ) {\text {PSL}}_2(\mathcal {O}) acts discontinuously on hyperbolic 3 3 -space H H . If Γ ≤ PSL 2 ( O ) \Gamma \leq {\text {PSL}}_2(\mathcal {O}) is a torsionfree subgroup of finite index then the manifold Γ ∖ H \Gamma \backslash H can be compactified to a manifold M Γ {M_\Gamma } so that the inclusion Γ ∖ H ≤ M Γ \Gamma \backslash H \leq {M_\Gamma } is a homotopy equivalence. M Γ {M_\Gamma } is a compact with boundary. The boundary being a union of finitely many 2 2 -tori. This paper contains a computer-aided study of subgroups of low index in PSL 2 ( O ) {\text {PSL}}_2(\mathcal {O}) for various O \mathcal {O} . The explicit description of these subgroups leads to a study of the homeomorphism types of the M Γ {M_\Gamma } .

Free quotients of congruence subgroups of Bianchi groups

Free quotients of congruence subgroups of Bianchi groups

Almost all Bianchi groups have free, non-cyclic quotients

Let d be a square-free positive integer and let O (= Od) be the ring of integers of the imaginary quadratic number field ℚ(-d). The groups PSL2(O) are called the Bianchi groups after Luigi Bianchi who made the first important contribution 1 to their study in 1892. Since then they have attracted considerable attention particularly during the last thirty years. Their importance stems primarily from their action as discrete groups of isometries on hyperbolic 3-space, H3. As a consequence they play an important role in hyperbolic geometry, low-dimensional topology together with the theory of discontinuous groups and automorphic forms. In addition they are of particular significance in the class of linear groups over Dedekind rings of arithmetic type. Serre9 has proved that in this class the Bianchi groups (along with, for example, the modular group, PSL2(z), where z is the ring of rational integers) have an exceptionally complicated (non-congruence) subgroup structure.

Classification of Maximal Fuchsian Subsgroups of Some Bianchi Groups

AbstractLet d = 1,2, or p, prime p ≡ 3 (mod 4). Let Od be the ring of integers of an imaginary quadratic field A complete classification of conjugacy classes of maximal non-elementary Fuchsian subgroups of PSL(2, Od) in PGL(2, Od) is given.

Parametrizing Fuchsian Subgroups of the Bianchi Groups

Let dbe a positive square-free integer and let Od denote the ring of integers in . The groups PSL2(Od) are collectively known as the Bianchi groups and have been widely studied from the viewpoints of group theory, number theory and low-dimensional topology. The interest of the present article is in geometric Fuchsian subgroups of the groups PSL2(Od). Clearly PSL2 is such a subgroup; however results of [18], [19] show that the Bianchi groups are rich in Fuchsian subgroups.

The euclidean bianchi group

The Bianchi Groups Γd are PSL2(0d) where 0d is the ring of integers in the quadratic imaginary number field Q√-d with d a positive square-free rational integer. If d=1,2,3,7,11 0d has a Euclidean algorithm and the corresponding groups are called the Euclidean Bianchi Groups. The group Γ1 is the Picard Group has been studied independently. Here the subgroup structure of the remaining Euclidean Bianchi Groups is investigated. We show that they have a unique normal subgroup of index in if(n,6)=1 among other results. We also classify the abelian and nilpotent subgroups and discuss the structure of both congruence and non-congruence subgroups.

B. Fine: Algebraic theory of the Bianchi groups

B. Fine: Algebraic theory of the Bianchi groups

Automorphisms in action: spatially homogeneous Einstein-Maxwell plane waves

The time-dependent automorphism formalism is applied in the computer algebra system SHEEP to identify the subclasses of Einstein-Maxwell plane-wave solutions which admit Bianchi groups acting on spacelike hypersurfaces. Using the classification program CLASSI, the invariant characterisations of these metrics are obtained and the results used to generalise previous work by Siklos (1984) on spatially homogeneous vacuum plane-wave cosmologies. The invariant characterisations are described through a three-dimensional phase space which physically represents the amplitude and polarisation of the plane wave and the electromagnetic field. A previous theorem on linearly polarised plane gravitational waves is also generalised.

The automorphism groups for Bianchi universe models and computer-aided invariant classification of metrics

An implementation of the time-dependent automorphism formalism for Bianchi cosmologies, in the computer algebra system SHEEP, is described. Its application to the particular case of Bianchi cosmologies admitting vacuum plane-wave solutions is given in relation to the equivalence problem. Using this approach, it is shown that the invariant characterisations of these metrics allow many properties of these spacetimes to be obtained in a straightforward manner. In particular, the invariant characterisations are used to determine the subclasses of spatially homogeneous vacuum plane-wave solutions which simultaneously admit distinct foliations with different Bianchi symmetries. The subclasses of the general vacuum plane-wave solution which admit a Bianchi group on a spacelike hypersurface are evaluated. These results are found to be expressible graphically, in the form of a phase plane which is physically interpretable as describing the amplitude and phase of polarisation of the plane wave. As a final application, it is shown that the Lifshitz-Khalatnikov Bianchi VIh solution is the only vacuum plane-wave solution for which the gravitational wave is linearly polarised.

Some Amalgam Structures for Bianchi Groups

Splittings as an amalgamated sum, where one of the factors is the projective elementary matrices, are displayed for all the Bianchi groups except for the Euclidian ones. Also splittings as an HNN group, where the base is ${\text {PS}}{{\text {L}}_2}(Z)$, are given for all the Bianchi groups except ${\text {PS}}{{\text {L}}_2}({O_1})$ and ${\text {PS}}{{\text {L}}_2}({O_3})$.

Some amalgam structures for Bianchi groups

Splittings as an amalgamated sum, where one of the factors is the projective elementary matrices, are displayed for all the Bianchi groups except for the Euclidian ones. Also splittings as an HNN group, where the base is PS L 2 ( Z ) {\text {PS}}{{\text {L}}_2}(Z) , are given for all the Bianchi groups except PS L 2 ( O 1 ) {\text {PS}}{{\text {L}}_2}({O_1}) and PS L 2 ( O 3 ) {\text {PS}}{{\text {L}}_2}({O_3}) .

Fuchsian Embeddings in the Bianchi Groups

If d is a positive square free integer we let Od be the ring of integers in and we let Γd = PSL2(Od), the group of linear fractional transformationsand entries from Od {if d = 1, ad – bc = ±1}. The Γd are called collectively the Bianchi groups and have been studied extensively both as abstract groups and in automorphic function theory {see references}. Of particular interest has been Γ1 – the Picard group. Group theoretically Γ1, is very similar to the classical modular group M = PSL2(Z) both in its total structure [4, 6], and in the structure of its congruence subgroups [8]. Where Γ1 and M differ greatly is in their action on the complex place C. M is Fuchsian and therefore acts discontinuously in the upper half-plane and every subgroup has the same property.