In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R).
Some Escher graphics are based on them (for the disc model of hyperbolic geometry).
is a model of the hyperbolic plane when endowed with the metric The group PSL(2,R) acts on
be a discrete group, which means that: Although discontinuity and discreteness are equivalent in this case, this is not generally true for the case of an arbitrary group of conformal homeomorphisms acting on the full Riemann sphere (as opposed to
Indeed, the Fuchsian group PSL(2,Z) is discrete but has accumulation points on the real number line
to every rational number, and the rationals Q are dense in R. A linear fractional transformation defined by a matrix from PSL(2,C) will preserve the Riemann sphere P1(C) = C ∪ ∞, but will send the upper-half plane H to some open disk Δ. Conjugating by such a transformation will send a discrete subgroup of PSL(2,R) to a discrete subgroup of PSL(2,C) preserving Δ.
The notion of an invariant proper subset Δ is important; the so-called Picard group PSL(2,Z[i]) is discrete but does not preserve any disk in the Riemann sphere.
Similarly, the idea that Δ is a proper subset of the region of discontinuity is important; when it is not, the subgroup is called a Kleinian group.
It is most usual to take the invariant domain Δ to be either the open unit disk or the upper half-plane.
Because of the discrete action, the orbit Γz of a point z in the upper half-plane under the action of Γ has no accumulation points in the upper half-plane.
This happens if the quotient space H/Γ has finite volume, but there are Fuchsian groups of the first kind of infinite covolume.
This is the subgroup of PSL(2,R) consisting of linear fractional transformations where a, b, c, d are integers.
Here Γ(n) consists of linear fractional transformations of the above form where the entries of the matrix are congruent to those of the identity matrix modulo n. A co-compact example is the (ordinary, rotational) (2,3,7) triangle group, containing the Fuchsian groups of the Klein quartic and of the Macbeath surface, as well as other Hurwitz groups.