Arboreal Group Theory: Proceedings of a Workshop Held by Juan M. Alonso (auth.), Roger C. Alperin (eds.)

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By Juan M. Alonso (auth.), Roger C. Alperin (eds.)

During the week of September thirteen, 1988 the Mathematical Sciences study Institute hosted a 4 day workshop on Arboreal crew idea. This quantity is the manufactured from that assembly. this system founded with regards to the idea of teams performing on timber and some of the purposes to hyperbolic geometry. subject matters contain the idea of size capabilities, constitution of teams appearing freely on bushes, areas of hyperbolic constructions and their compactifications, and moduli for tree actions.

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X1lwnx2n+l ... }) U ({X2n ... }\{XZ;+lW;;-lxl ... M. Alonso where Xl1wnx2n+t denotes X2X3 ... X2~ , and similarly for its inverse. Write (26) By (a) above, the inclusions of K are admissible with transversals, say, Tl in the first factor, and T2 in the second. We have seen in the proof of (a) that T2 can be taken to be the set of all words beginning with W n , or xt~+l' or else with xt~ but excluding those that begin with [X2n, X2n+l]. Clearly, the inclusion of A in G 2 is length preserving.

Let e = eG u , where e = apv-Ih- I . It follows e-Ia = hvp-l, e-lb = hvfq = = hVPAc(A)AAt = PGE(C)A:::'AAt, and hence d(e, a) = d(e,b) = ie(B)1 and d(e,c) = ie(C)I. Consequently, e E Y(a,b,c), as PBE(B)A~q, e-le Ihl, required. It remains to observe that d(sa,sb) the group 7r acts on the A-tree X. 2. The pair (7rI (X, T), XT) does not depend, up to an isomorphism, on the choice of T. PROOF: Let T' = (X, (l~y)(x,Y)Ex2) be another subpretree of X and Xo be a vertex of X. 5, and let Fl : 7rl (X, T) ~ 7rl (X, T') be the composite isomorphism Tli;,xo 0 TlT,xo' Identifying G x , x E X, with a common subgroup and E(x,y),x, y EX, with a common subset of 7rl (X, T) and 7rl (X, T'), we get Fl (s) = 1xoxs1xxo and FI(f) = 1xoxflyxo for s E C x , f E E(x,y).

For g E E(y, z), let [g] = {(j,p) : fEE, t(j) = y, pEG J\G y and 8(j,p,g) = O}, and O:g : [g] ~ [O,lgl] be the map given by O:g(j,p) = If I· For (j,s,g) E E(x,y) x G y x E(y,z), let Y(j,s,g) = ((h,t) E [f] : 8(h,tp(j,s,g), c(j,s,g» = 8(c(h,t,j), p(/,BjlrI,h)-ls,g) = O}. [g], O:g and Y(j, s, g) are well-defined according to the remark above. The following statements are equivalent: X is a A-graph of groups. X has the next properties: O:g is bijective for each gEE; Y(j,s,g) is non-empty for each suitable triple (j,s,g).

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