## $$L_\delta$$ 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 $$L_\delta$$.

### MSC:

 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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