# Schottky group

In mathematics, a **Schottky group** is a special sort of Kleinian group, first studied by Template:Harvs.

## Definition

Fix some point *p* on the Riemann sphere. Each Jordan curve not passing through *p*
divides the Riemann sphere into two pieces, and we call the piece containing *p* the "exterior" of the curve, and the other piece its "interior".
Suppose there are 2*g* disjoint Jordan curves *A*_{1}, *B*_{1},..., *A*_{g}, *B*_{g} in the Riemann sphere with disjoint interiors.
If there are Möbius transformations *T*_{i} taking the outside of *A*_{i} onto the inside of *B*_{i}, then the group generated by these transformations is a Kleinian group. A **Schottky group** is any Kleinian group that can be constructed like this.

## Properties

Schottky groups are finitely generated free groups such that all non-trivial elements are loxodromic. Conversely Template:Harvtxt showed that any finitely generated free Kleinian group such that all non-trivial elements are loxodromic is a Schottky group.

A fundamental domain for the action of a Schottky group *G* on its regular points Ω(*G*) in the Riemann sphere is given by the exterior of the Jordan curves defining it. The corresponding quotient space Ω(*G*)/*G* is given by joining up the Jordan curves in pairs, so is a compact Riemann surface of genus *g*. This is the boundary of the 3-manifold given by taking the quotient (*H*∪Ω(*G*))/*G* of 3-dimensional hyperbolic *H* space plus the regular set Ω(*G*) by the Schottky group *G*, which is a handlebody of genus *g*. Conversely any compact Riemann surface of genus *g* can be obtained from some Schottky group of genus *g*.

## Classical and non-classical Schottky groups

A Schottky group is called **classical** if all the disjoint Jordan curves corresponding to some set of generators can be chosen to be circles. Template:Harvs gave an indirect and non-constructive proof of the existence of non-classical Schottky groups, and Template:Harvtxt gave an explicit example of one. It has been shown by Template:Harvtxt that all finitely generated classical Schottky groups have limit sets of Hausdorff dimension bounded above strictly by a universal constant less than 2. Conversely, Template:Harvtxt has proved that there exists a universal lower bound on the Hausdorff dimension of limit sets of all non-classical Schottky groups.

## Limit sets of Schottky groups

The limit set of a Schottky group, the complement of Ω(*G*), always has Lebesgue measure zero, but can have positive *d*-dimensional Hausdorff measure for *d* < 2. It is perfect and nowhere dense with positive logarithmic capacity.

The statement on Lebesgue measures follows for classical Schottky groups from the existence of the Poincaré series

Poincaré showed that the series | *c*_{i} |^{–4} is summable over the non-identity elements of the group. In fact taking a closed disk in the interior of the fundamental domain, its images under different group elements are disjoint and contained in a fixed disk about 0. So the sums of the areas is finite. By the changes of variables formula, the area is greater than a constant times | *c*_{i} |^{–4}.^{[1]}

A similar argument implies that the limit set has Lebesgue measure zero.^{[2]} For it is contained in the complement of union of the images of the fundamental region by group elements with word length bounded by *n*. This is a finite union of circles so has finite area. That area is bounded above by a constant times the contribution to the Poincaré sum of elements of word length *n*, so decreases to 0.

## Schottky space

Schottky space (of some genus *g* ≥ 2) is the space of marked Schottky groups of genus *g*, in other words the space of sets of *g* elements of PSL_{2}(**C**) that generate a Schottky group, up to equivalence under Moebius transformations Template:Harv. It is a complex manifold of complex dimension 3*g*−3. It contains classical Schottky space as the subset corresponding to classical Schottky groups.

Schottky space of genus *g* is not simply connected in general, but its universal covering space can be identified with Teichmüller space of compact genus *g* Riemann surfaces.

## See also

## Notes

## References

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