By Buekenhout F., Huybrechts C.

We end up the lifestyles of a rank 3 geometry admitting the Hall-Janko workforce J2 as flag-transitive automorphism crew and Aut(J2) as complete automorphism workforce. This geometry belongs to the diagram (c·L*) and its nontrivial residues are whole graphs of measurement 10 and twin Hermitian unitals of order three.

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**Additional info for A (c. L* )-Geometry for the Sporadic Group J2**

**Example text**

These co–ordinates are actually maps from subsets of the manifold R2 to the “standard” Euclidean space, which in this case also happens to be R2 . The first chart U1 can be taken to be all of R2 , and the function φ1 is just the identity φ1 : (x, y) → (x, y). The second function φ2 is not defined on all of R2 , and might be given in the chart U2 = {x, y > 1} for instance, by x x2 + y 2 , arctan φ2 : (x, y) → y Defining ρ = φ2 φ−1 1 = φ2 as above, check that dρ is invertible everywhere in the overlap of the two charts.

What about if Σ is a punctured surface of genus g ≥ 2? 5. Dehn twists and Lickorish’s theorem. 59. An oriented (polyhedral) simple closed curve c in a surface Σ and an annulus neighborhood A of c parameterized as S 1 ×I define a homeomorphism tc : Σ → Σ by x→x for x outside A tc : (θ, t) → (θ − 2tπ, t) for (θ, t) ∈ A This homeomorphism is known as a Dehn twist about c. As an element of MC(Σ), it depends only on the isotopy class of c. Note that [tc ]−1 = [tc ] where c denotes c with the opposite orientation.

If αi is the common loop in ∂P1 ∩ ∂P2 , then βi is a dual curve which cuts Q into two other pairs of pants P1 , P2 such that P1 has one boundary component in common with each of P1 , P2 and similarly for P2 . Then twisting αi through 2π replaces βi with a new curve tαi (βi ), the curve obtained by a Dehn twist around αi . Briefly: βi decomposes into two arcs δ, along αi , and αi decomposes into two arcs µ, ν along βi . Then tαi (βi ) is the simple closed curve homotopic to δ∗µ∗ ∗ν, wher ∗ denotes concatenation of arcs.