Geodesic
On Subsemigroups of the Projective Group on the Line
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 1001-1011

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The subsemigroups of the projective group on the line that are described in this paper are those that can be generated by a pair of infinitesimal transformations. One denotes by G the connected component of the identity of this group; Theorem 1 gives a necessary and sufficient condition for a pair of infinitesimal transformations to generate a subsemigroup which is equal to G (and hence is actually a group). This condition is reformulated in a geometric manner in Theorem 1*. 
Lowenthal, Franklin. On Subsemigroups of the Projective Group on the Line. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 1001-1011. doi: 10.4153/CJM-1968-097-7
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[1] 1. In (2), the solution can be put into the form y = (ax + b)/(cx + d), where a, b, c, and d are all real and ad - be 0.

[2] 2. Assume, for definiteness, that the sink interval is just (zi, W\) which can be achieved by inner automorphism if necessary.

[3] 3. Observe that this is clearly a product of length 3. As the order of applying transformations always begins at the right, parentheses will be omitted subsequently.

[4] 4. The parameters t, s will be used from now on.

[5] 5. In fact, ϒ = V(T2), ϒ ≧ l , but this is not needed. It shows that ϒ (τ)is monotonically increasing, a fact which subsequently is not used.

[6] 6. If ॉ is elliptic, and ॉ and n have a common root, then they generate the same subgroup.

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