1. A Portrait of a Group on a Surface with Boundary
Draw a picture of a finite group on a compact orientable surface with one or more boundary components.
A bordered Klein surface is a compact surface with one or more open disks deleted from the surface.

2. Labeling by Group Elements
The elements of the group are represented by transformations of the surface.
Each group element takes a “fundamental region” to another region and the image of the fundamental region under this element is labeled by the element.
Regions are distinguished by whether the orientation is preserved or reversed and colored differently.

3. Transformations of the Plane
The surface is formed by identifying regions of a hyperbolic plane. We use the Poincare Disk model of the Hyperbolic plane.
Burnside used this approach in 1911 with inversions in circles as the transformations of the plane, which correspond to transformations of the Hyperbolic plane.

4. Regions in the Plane
Inversions in a circle always reverse orientation.
Each region is bounded by the fixed points of such an inversion.
This means that adjacent regions always have a different color.
These regions tile the compact surface.

5. Free Groups
This gives a picture of the Free Group on two generators as a “Triangle group” in the Hyperbolic Plane.

7. Finite Groups
A finite group is obtained from such a free group by identifying words that represent the same group element.
This corresponds to identifying the regions in the hyperbolic plane labeled with words representing the same group element.
This identification gives the compact surface.

8. The Quaternion Group of Order 8

9. Bordered Klein Surfaces
In order to model a boundary region, we must identify a region and an adjacent region.
This means that one of the inversions must be identified with the identity.
Mathematically, one inversion is in the Kernel of the homomorphism from the infinite group to the finite group.

10. The Homomorphism
This map is a homomorphism from a Non-Euclidean Crystallographic group (NEC – group)  to the finite group.
It follows that  has at least 4 generators.
This group  with maximal symmetry is generated by 4 involutions; a, b, c and d with additional relations (ab)2 = (bc)2 = (cd)2 = (da)3 = 1.

11. Image in the Hyperbolic Plane
Each region in the Hyperbolic Plane has at least FOUR sides instead of three.
Each face is bounded by curves of different colors, each representing a different involution.
One or more of these curves borders a boundary component.

12. The Real Genus
The algebraic genus of a compact orientable surface X of topological genus p with k holes is given by g(X) = 2p + k ̶ 1.
The real genus of the group G is the number ρ(G) which is the smallest g(X) for any bordered surface X whose group of isometries contains a group isomorphic to X.

13. Constraints |G| ≤ 12(ρ(G) – 1) by Hurwitz Theorem.
A group that satisfies this bound has maximal symmetry and is called an M* group.
There are only two solvable M* groups, the symmetric group S4 and the dihedral group D6.

14. The Groups
The group S4 has real genus 3 and acts on a sphere with 4 holes (this is topologically equivalent to a tetrahedron with a hole at each vertex).
Since S4 is the symmetry group of the tetrahedron, we can construct the regions without all of this theory.

15. Action of S4 on a Tetrahedron

16. Stereographic Projection of the Portrait of S4

17. The M* group Z2  S3  D6
The dihedral group D6 of order 12 has real genus 2 and is an M* group.
It can act on a sphere with three holes or on a torus with one hole.
Both actions occur and result in interesting portraits.

18. A Sphere with Three Holes.

19. The Torus with One Hole

20. Modified Homomorphism
These finite M* - groups are also images of a group generated by three involutions t, u and v satisfying (tu)2 = (tv)3 = 1.
It follows that all vertices in a portrait of an M* - group, that are not on the boundary, must have degree 4 or degree 6.
The order of the image of uv is called the “action index” of the group.

21. Consequences
The order of uv determines how many faces are adjacent to each boundary component.
Every region must border one of the boundary components.
All boundary components look alike.

22. Polygonal Representation of S4 on a sphere with 4 Holes

23. Polygonal representation of D6 as a Torus with One Hole

24. Group with Real Genus 5
Let G be a group generated by three involutions t, u and v satisfying (tu)2 = (tv)2 = (cd)4 = (uv)8 = 1 and (uv)3 = tvut.
This group has order 32 and it is the group of automorphisms of the regular map {4,8}1,1.
Also |G| = 8(ρ(G) – 1) and so it is NOT an M* - group.

25. Surface Characteristics
The group G acts on a double torus with two holes.
The regular map {4,8}1,1 has computer graphics already on the web, so I have not constructed a physical model of it.

26. Polygonal Representation

27. G on a Double Torus

28. THE END