The term double torus is occasionally used to denote a genus 2 surface.

We are interested to understand the **structure **of a group G (is it abelian, or a product of abelian groups, or a free group of some kind etc.). In general, this is a (very) hard question. One strategy is to let G **act **on some well known (topological) space. Often by studying the **effect of this group action **on the space, it is possible to comment on the algebraic structure of the group.

Why this ‘indirect’ method should be regarded as ‘natural’? It is useful to think about elements of groups as ‘actors’. Their true color is revealed, only when they are unleashed in a stage (topological space).

However, we have to find the **right **space on which the **group under scanner **is to be let loose. A lot of effort goes into the construction (and investigation) of topological spaces which will be effective stages for group action.

Bass Serre theory produced a wonderful (topological) space that produces important information about groups acting on them. They are cleverly designed simplicial trees. We look at the stabilizers of vertices and edges of this tree. This process reveals a lot of information about the **structure **of the group.

Simplicial trees are different from (non-simplicial) R-trees (real trees). The key distinction is: R-trees have non-discrete branching points. There is an interesting construction in this context. Let **G **act isometrically (preserving distances), on a sequence of **negatively curved spaces. Then **we have a natural isometric action of G on an R-tree **in the Gromov Hausdorff limit. **

This makes R-trees the **final destination space **of the isometric Group action. It makes sense to **study the R-trees. **After all, they are stage of action of the Group that is our ultimate object of interest. A process of resolution leads us from R-trees to (measured) laminated 2-complexes.

(**Ref: Bestvina; 1999)**

(Image: https://conan777.wordpress.com/tag/measured-lamination/)

Though I am B.Tech (Mechanical Engineering) graduate having interest in Mathematics. I had taken Mathematical Engineering subjects like Boundary Value Problems;Elasticity:Fluid Mechanics; Mechanics of Solids;Vibrations;Engineering Mathematics ;Calculus of Variations etc-etc.

Things mentioned above by you is bit of pure Mathematics.I appreciate the beauties In diagram but I have To do some serious beginning for this in my leisure hours as I have not made any systematic study of them.

Anyway Very Impressive Illustration by you.

Debjyoti, some of the most interesting mathematical innovations were part of engineering investigations.