The loop shortening property and almost convexity. (Q1421745)

Let \(G\) be a group equipped with a finite generating set \(X\) and let \(\Gamma\) be the corresponding Cayley graph. Two paths (respectively loops) \(w\) and \(u\) in \(\Gamma\) are said to \(k\)-fellow travel if \(d(w(t),u(t))\leq k\) for all \(t\geq 0\), and they are said to asynchronously \(k\)-fellow travel if there is a proper monotone increasing function \(\phi:[0,\infty[\to [0,\infty[\) such that \(d(w(\phi(t)),u(\phi(t)))\leq k\) for all \(t\geq 0\). The pair \((G,X)\) is said to have the (asynchronous) falsification by fellow traveler property if there exists \(k>0\) such that for each nongeodesic path \(w\), there is a path \(u\) with the same endpoints such that \(| u| <| w|\) and \(w\),\(u\) (asynchronously) \(k\)-fellow travel. (This notion is due to Neumann and Shapiro.) The author then introduces the following two notions : The pair \((G,X)\) is said to have the (asynchronous) loop shortening property if there exists \(k>0\) such that for any loop \(w\), there exists a loop \(u\) such that \(| u| <| w|\) and \(w\),\(u\) (asynchronously) \(k\)-fellow travel. The pair \((G,X)\) is said to have the (asynchronous) basepoint loop shortening property if there exists \(k>0\) such that for any loop \(w\) based at \(w(0)\), there exists a loop \(u\) based at \(w(0)\) such that \(| u| <| w|\) and \(w\),\(u\) (asynchronously) \(k\)-fellow travel. Finally, the author uses Cannon's notion of almost convexity: \((G,X) \) is said to be almost convex if there exists a constant \(C\) such that every pair of points lying at distance at most two apart and within distance \(N\) of the identity in \(\Gamma\) are connected by a path of length at most \(C\) which lies within distance \(N\) of the identity. The property of being almost convex depends on the choice of the generating set. The author proves the following results: Theorem: The pair \((G,X)\) has the asynchronous (basepoint) loop shortening property if and only if it has the synchronous (basepoint) loop shortening property. Theorem: If \((G,X)\) has the loop shortening property then \(G\) is finitely presented, and has a quadratic isoperimetric function. Theorem: If \((G,X)\) has the basepoint loop shortening property then it is almost convex. Finally, the author studies a certain class of multiple HNN extensions, and he then considers four particular group presentations which exhibit several kinds of properties involving the various notions mentioned above, as well as the property of admitting a quadratic isoperimetric Dehn function.