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Isoperimetric inequalities and the homology of groups. (English) Zbl 0829.20053

Let \(G=F/R\) be a finitely presented group, with \(F\) being a finitely generated free group and \(R\) the normal closure of a finite set of relators. For every word \(w\) corresponding to an element of \(R\), the area of \(w\) is the least number of relators needed to prove that \(w\) is the identity in \(G\). The isoperimetric function \(\Phi_G(n)\) is defined as the maximum of the areas of all words in \(R\). The authors define two other functions, the centralized isoperimetric function \(\Phi^{\text{cent}}_G\), whose definition uses the image of \(w\) in \(R/[R,F]\), and the abelianized isoperimetric function \(\Phi^{\text{ab}}_G\) whose definition uses the module \(R/[R,R]\). The complexity of \(\Phi^{\text{cent}}_G\) is essentially that of the integral homology \(H_2 G\) and therefore the authors can use algebraic tools for studying it. There is an order relation \(\preceq\) on such functions, where \(f\preceq g\) means that there exist integers \(K_1\), \(K_2\) and \(K_3\) such that \(f(n)\leq K_2 g(K_1 n)+K_3n\) for all \(n\). One has the relations \(\Phi^{\text{cent}}_G(n)\preceq\Phi^{\text{ab}}_G(n)\preceq\Phi_G(n)\). The generated equivalence relation is denoted by \(\simeq\). It is well-known that quasi-isometric groups have \(\simeq\)- equivalent isoperimetric functions \(\Phi\). The authors prove the same result for \(\Phi^{\text{ab}}_G\), and they prove that the result fails for \(\Phi^{\text{cent}}_G\). They establish new lower bounds for the functions \(\Phi\), \(\Phi_G^{\text{cent}}\) and \(\Phi^{\text{ab}}_G\), and they provide examples of finitely generated groups with isoperimetric functions of special type.
The main results are the following: Let \(N\) be a free nilpotent group of class \(c\) and finite rank \(\geq 2\). Then \(n^{c+1}\preceq\Phi^{\text{cent}}_N (n)\). For every integer \(e\), there exists a finitely presented group \(M\) with \(\Phi_M(n)\simeq\Phi^{\text{cent}}_M(n)\simeq n^e\). (The authors note that methods of M. Bridson and C. Pittet can be used to improve this result.) There exists a metabelian polycyclic group \(G\) of Hirsch length 3 such that \(\Phi_G(n)\simeq\Phi^{\text{cent}}_G(n)\simeq 2^n\). (S. Gersten had already shown that a nilpotent group of Hirsch type \(h\) has an isoperimetric function of order \(\preceq n^{2^h}\).) For a finitely generated nilpotent group \(G\) of Hirsch length \(h\geq 2\), one has \(\text{rk}_Z(H_h G)=1\) and \({h\choose i}\geq\text{rk}_Z(H_i G)\geq 2\) for \(i=1,\dots,h-1\). The authors give a general prescription for constructing groups \(G\) such that \(\Phi^{\text{cent}}_G\) is \(\preceq\) any preassigned function. There is a set of examples introduced by W. Thurston of nilpotent groups of class 2 for which the authors prove that \(\Phi^{\text{cent}}(n)\simeq n^2\). (Thurston had announced the stronger result that \(\Phi(n)\simeq n^2\).) The authors then use the free differential calculus to obtain lower bounds for \(\Phi^{ab}\). They obtain in particular lower bounds (already obtained by Gersten) for several one relator groups. Using this calculus, they prove also the following new results: Let \(G\) be the group \(\langle a_1,\dots,a_s,b_1,\dots,b_t\mid u=v\rangle\), with \(u\) a word on the \(a_i\) and \(v\) a word on the \(b_j\). Then, \(\Phi_G(n)\preceq n^2\). Let \(G=\langle a_1,\dots,a_s,b_1,\dots,b_t\mid u^p=v^q\rangle\), with \(u\) a word on the \(a_i\) and \(v\) a word on the \(b_j\) and \(p,q>1\). Then, \(\Phi_G(n)\simeq\Phi^{\text{ab}}_G(n)\simeq n^2\). The paper ends with the following general open problem: determine \(\Phi^{\text{cent}}_G\) where \(G\) is a torsion-free one relator group.

MSC:

20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
57M05 Fundamental group, presentations, free differential calculus
20F18 Nilpotent groups
20F19 Generalizations of solvable and nilpotent groups
20F16 Solvable groups, supersolvable groups
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References:

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