\(L_\delta\) groups are almost convex and have a sub-cubic Dehn function. (English) Zbl 1056.20030

The author introdues some definitions for a metric property \(L_\delta\) of groups, almost convex group with finite generating set \(\widehat g\), and Dehn function \(D(n)\) (isoperimetric function) for a group: (i) For any \(\delta\geq 0\) and finite sequence of points \(x_1,x_2,\dots,x_n\), the \(n\)-tuple \((x_1,x_2,\dots,x_n)\) is a ‘\(\delta\)-path’ if \(d(x_1,x_2)+\cdots+d(x_{n-1},x_n)\leq d(x_1,x_n)+\delta\). (ii) For any \(\delta\geq 0\) the space \(X\) has ‘property \(L_\delta\)’ if for each three distinct points \(x,y,z\in X\) there exists a point \(t\in X\) so that the paths \((x,t,y)\), \((y,t,z)\) and \((z,t,x)\) are all \(\delta\)-paths. (iii) A group \(G\) with finite generating set \(\widehat g\) is ‘almost convex’ if there is a constant \(C\geq 0\) so that for any two vertices that lie distance \(n\geq 0\) from the identity vertex and at most 2 apart from each other, there is a path connecting them that lies inside the closed ball of radius \(n\) and has length at most \(C\). (iv) The ‘Dehn function’ for \(\langle\widehat g\mid R\rangle\) is \(D(n)=\max\{A(w):w\) has at most \(n\) letters, and \(w\) evaluates to the identity}. An ‘isoperimetric function’ for \(\langle\widehat g\mid R\rangle\) is any function which satisfies \(f(n)\geq D(n)\).
The main results of the paper are the following theorems: if a Cayley graph for a group \(G\) has the property \(L_\delta\) for some \(\delta\geq 0\) then it is almost convex with constant \(3\delta+2\), and that if the Cayley graph for a group \(G\) with respect to some finite generating set \(\widehat g\) has the property \(L_\delta\) for some \(\delta\geq 0\) then \(G\) is finitely presented, and has an isoperimetric function equivalent to \(n^{1/(1-\log_32)}\). The paper ends with an open question: If there exists a group with an isoperimetric function greater than quadratic and less than the sub-cubic bound given in the theorem which has \(L_\delta\).


20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
20F05 Generators, relations, and presentations of groups
57M07 Topological methods in group theory
20F06 Cancellation theory of groups; application of van Kampen diagrams
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