By Karl H. Hofmann
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Extra info for Lectures on Rings and Modules
2) Indeed, the label of the top path of the connecting a-band commutes with all letters in A(P1 ). 10 (see also part (iii) of the definition of non-reduced diagrams). 32 (3) Proving by contradiction, consider the subdiagram Γ bounded by ∂(P i) and T. 11, that Γ has no cells except for G-cells, and that |T| = 0. If Γ contains a G-cell, it can be merged with Π. If Γ does not contain cells, we obtain a contradiction with the assumption that the boundary label of Π is cyclically reduced. 6 Shifting indexes The following lemma utilizes the fact that our S-machines work the same way in all parts of words Σr,i (w1 , w2 , w3 , w4 ) between two consecutive Kj -letters.
Hence the base of the label of the bottom p4 will be called the base of the trapezium (it is equal to the base of each of Ti ). Notice that every k-band is a trapezium with a 1-letter base. A reduced annular diagram ∆ is called a ring with boundary components p2 and p4 if: (1) the labels of p2 and p4 contain no θ-edges; (2) the boundaries p2 and p4 are minimal; (3) ∆ has at least one (k, θ)-cell. It follows from the definition, that a ring ∆ contains at least one k-band having a (θ, k)-cell, every maximal k-band of ∆ connects p2 and p4 , and a ring is obtained from a trapezium by identifying two bands B and B (in this case B must be a copy of B in the definition of trapezium).
Vd−1 )a ≡ (Ud )a . Also every a-band which starts on bot(T1 ) ends on p4 , every a-band which starts on top(Td ) ends on p2 . If an a-band starts and ends on p4 (resp. p2 ) then the label of the a-path between the start and end edges of that band must be freely equal to 1. Hence (V0 )a = (U1 )a and (Vd )a = (Ud+1 )a in the free group. (2) Cells in Γi do not contain θ-edges. 8) or G-relations. 8), the relations become trivial. , d, that diagram becomes a diagram with G-cells only. ) Since the portions of p1 and p3 on the contour of Γi consist of X-edges, these portions collapse to vertices.
Lectures on Rings and Modules by Karl H. Hofmann