Bianchi group

In mathematics, a Bianchi group is a group of the form In mathematics, a Bianchi group is a group of the form where d is a positive square-free integer. Here, PSL denotes the projective special linear group and O d {displaystyle {mathcal {O}}_{d}} is the ring of integers of the imaginary quadratic field Q ( − d ) {displaystyle mathbb {Q} ({sqrt {-d}})} . The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of P S L 2 ( C ) {displaystyle PSL_{2}(mathbb {C} )} , now termed Kleinian groups. As a subgroup of P S L 2 ( C ) {displaystyle PSL_{2}(mathbb {C} )} , a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space H 3 {displaystyle mathbb {H} ^{3}} . The quotient space M d = P S L 2 ( O d ) ∖ H 3 {displaystyle M_{d}=PSL_{2}({mathcal {O}}_{d})ackslash mathbb {H} ^{3}} is a non-compact, hyperbolic 3-fold with finite volume, which is also called Bianchi manifold. An exact formula for the volume, in terms of the Dedekind zeta function of the base field Q ( − d ) {displaystyle mathbb {Q} ({sqrt {-d}})} , was computed by Humbert as follows. Let D {displaystyle D} be the discriminant of Q ( − d ) {displaystyle mathbb {Q} ({sqrt {-d}})} , and Γ = S L 2 ( O d ) {displaystyle Gamma =SL_{2}({mathcal {O}}_{d})} , the discontinuous action on H {displaystyle {mathcal {H}}} , then The set of cusps of M d {displaystyle M_{d}} is in bijection with the class group of Q ( − d ) {displaystyle mathbb {Q} ({sqrt {-d}})} . It is well known that any non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.