By Xiangdong Xie

We determine a Bowen variety pressure theorem for the elemental staff of a noncompacthyperbolic manifold of finite quantity (with measurement a minimum of 3).

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2 (Enochs [I]). If a torsion-complete p-group A is contained in the direct sum C = Q i C iof separable p-groups C i ,then there exist an integer m and afinite number C,,, . . , C,, of the C i such that pmA[p]5 C , , 0 . a A satisfactory description of direct decompositions of torsion-complete groups is given in the next theorem. 3 (Kulikov [2]). r f a torsion-complete p-group A is the direct sum of infinitely many subgroups A i , then the A , are torsion-complete and for a suflciently large integer m, pmAi= 0 for almost all i.

Furthermore, G as a summand of a pure-complete Obviously, G[p] = group is likewise pure-complete. 3) completes the proof. 0 urn. The socles play a crucial role in direct sums of torsion-complete groups. Convincing evidence is furnished by the next theorem. 5 (Hill [ 5 ] ) . Lei A = O i A i and G = @ j E J Gj be pure subgroups of some p-group H such that A [ p ] = G [ p ] , where each of A i , Gj is torsion-complete. Then A z G. A simple consequence of (71. I ) is that, for every i, there exist a decomposition A i = E i 0C iand a finite subset J ( i ) of J such that Ei is bounded and Ci[p] 5 0, E J ( i ) G, .

Let T be a separable p-group with basic subgroup B such that B/T is of finite rank r . Show that Pext(Z(p"), T ) = @ J , . ] 15. If T is as in Ex. 14 and if S is a separable p-group which contains T as a pure dense subgroup such that r ( S / T )= r, then S z B. ] 69. FURTHER CHARACTERIZATIONS OF TORSION-COMPLETE p-GROUPS In the preceding section, the torsion-complete p-groups were characterized in various ways in terms of algebraic properties involving other groups, too. Here our main objective will be to obtain intrinsic algebraic characterizations for them.