# Coordonnées parallèles et axiales: Méthode de transformation by Maurice d'Ocagne

By Maurice d'Ocagne

This quantity is made from electronic pictures from the Cornell college Library old arithmetic Monographs assortment.

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In the previous example, the level sets are trees, and so the kernel is a free group. In examples of later sections, we shall encounter situations where the kernel may be ﬁnitely generated but not ﬁnitely presentable, or ﬁnitely presented but not of type F3 . Example (Morse Criterion Examples). Run the Morse criterion on the following group presentations. Say what the rank of the free kernel is. Write out the automorphism of the free group explicitly. 1. a, b | aba = bab 2. a, b | abaa = bb 3. a0 , .

3 (Circle-valued Morse function). A circle-valued Morse function on an aﬃne cell complex X is a cellular map f : X → S 1 , with the property that f lifts to a Morse function between universal covers. 4 (Ascending and descending links). Suppose X is an aﬃne cell complex and f : X → S 1 is a circle-valued Morse function. Choose an orientation of S 1 , which lifts to one of R, and lift f to a map of universal covers f : X → R. Let v ∈ X (0) , and note that the link of v in X is naturally isomorphic to the link of any lift v˜ of v in X.

Are all Z subgroups of these groups undistorted? Do these types of examples embed into non-positively curved groups such as the CAT(0) groups introduced in the next chapter? 4 (Higher dimensional versions of Dehn functions). Let r be a positive integer. The snowﬂake groups Gr,P admit monomorphisms φr which take the generators of the vertex groups to their rth powers, and which ﬁx all the stable letters si . The suspended snowﬂake group ΣGr,P is deﬁned to be the multiple (2fold) ascending HNN extension of Gr,P where each of the stable letters acts via φr .