Bianchi group
Encyclopedia
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 Od is the ring of integers of the imaginary quadratic field Q(√−d).
The groups were first studied by Luigi Bianchi
in the 1890s as a natural class of discrete subgroups
of PSL2(C).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Bianchi group is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of the form
- PSL2(Od)
where d is a positive square-free integer
Square-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
. Here, PSL denotes the projective special linear group and Od is the ring of integers of the imaginary quadratic field Q(√−d).
The groups were first studied by Luigi Bianchi
Luigi Bianchi
- External links :* offers translations of some of Bianchi's papers, plus a biography of Bianchi.* PDF copy at * * * *...
in the 1890s as a natural class of discrete subgroups
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
of PSL2(C).
External links
- Allen Hatcher, Bianchi Orbifolds