Cohomological Characterization of Hyperbolic Groups
نویسنده
چکیده
The nitely presented group G is hyperbolic ii H 2 (1) (G; ` 1) = 0. I have established the following result, full details for which will appear in a forthcoming paper. Theorem 1. The nitely presented group G is hyperbolic ii H 2 (1) (G; ` 1) = 0. For the deenition of thè 1-cohomology, see Ge1]. The coeecient group`1 here is the Banach space of bounded functions on a countably innnite set in the sup norm. One direction in the theorem, that H 2 (1) (G; ` 1) = 0 if G is hyperbolic, is established in Ge1]. It comes down, after a series of reductions, to the Hahn-Banach extension theorem. Another proof of this direction based on a diierent principle is contained in Ge2]. The proof of the converse result is based on the following Theorem 2. Suppose that P is a nite presentation for the group G which is not hyperbolic. Then there exists K > 0 such that for innnitely many t ! 1 there exist simple circuits w(t) in the Cayley graph ? so that (1) `(w(t)) 100t, and (2) for all n > 0, Area(w(t) n) Knt 2. Explanation. Area(w) is the integral lling norm N Z of the 1-cycle determined by w, or equivalently Area(w) is the minimal area of an oriented surface diagram lling the 1-cycle w in ?. This is less than the minimal area Area 0 (w) of a van Kampen (i.e. genus 0) lling of w, in general. For the deenitions of the integral lling norm N Z and of the real lling norm N R which we need below, see Ge1]. The circuit w(t) n goes n-times around the original circuit w(t). Since we shall only be considering these as 1-cycles in ?, the choice of the base point is irrelevant. Corollary 3. Under the hypotheses of Theorem 2 the real lling norm N R is not equivalent to thè 1 norm on integral 1-cycles on ?. Proof. We argue by contradiction, assuming to the contrary that N R j j 1 on integral 1-cycles. Using methods of Ge1] x9 I show there exists C > 0 so that for every integral 1-cycle b 6 = 0 on ? there exists an integer n 1 > 1 so that N Z (n 1 b)
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تاریخ انتشار 1996