A note on the Poénaru condition (Q2776805)

Let \(G\) be a group generated by a finite set \(X\). Let \(\Gamma(G,X)\) be the Cayley graph of \(G\) with respect to \(X\) and let \(d_X\) denote the word-metric on \(\Gamma(G,X)\).NEWLINENEWLINENEWLINEFor an integer \(n\geq 1\) the group \(G\) is said to satisfy the Poénaru condition \(P(n)\) with respect to the generating set \(X\) provided that there exists a function \(f:\mathbb{Z}_+\to\mathbb{R}_+\) such that the following holds: (1) For any number \(A>0\) we have \(\lim_{k\to\infty}k-Af(k)=\infty\). (2) Suppose that \(a,b\in G\) are elements at distance \(k\) from the identity element of \(G\) in the Cayley graph \(\Gamma(G,X)\) such that \(d_X(a,b)\leq n\). Then there is a path \(\alpha\) of length at most \(f(k)\) from \(a\) to \(b\) in the Cayley graph \(\Gamma(G,X)\) such that \(\alpha\) is contained in the closed ball of radius \(k\) around 1 in \(\Gamma(G,X)\).NEWLINENEWLINENEWLINEThis notion was suggested by \textit{V. Poénaru} [in J. Differ. Geom. 35, No. 1, 103-130 (1992; Zbl 0777.57001)] and further developed by \textit{V. Poénaru} [in Topology 33, No. 1, 181-196 (1994; Zbl 0830.57010)] as well as by \textit{V. Poénaru} and \textit{C. Tanasi} [in Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 20, No. 3, 387-414 (1993; Zbl 0821.57014)]. Poénaru formulated this condition only for functions of type \(f(k)=Ak^t+B\) where \(0\leq t<1\). The above definition was first explicitly stated by \textit{L. Funar} [in Arch. Math. 72, No. 2, 81-85 (1999; Zbl 0932.57003)].NEWLINENEWLINENEWLINEThe author defines the following conditions: Let \(n\geq 2\) be an integer. The group \(G\) is said to satisfy condition \(K(n)\) if there is \(k_0\geq 1\) with the following property. Suppose that \(k\geq k_0\) and \(a,b\in G\) are elements at distance \(k\) from the identity element of \(G\) in the Cayley graph \(\Gamma(G,X)\) such that \(d_X(a,b)\leq n\). Then there is a path \(\alpha\) of length less than \(2k\) from \(a\) to \(b\) in \(\Gamma(G,X)\) such that \(\alpha\) is contained in the closed ball of radius \(k\) around 1 in \(\Gamma(G,X)\). Similarly condition \(K'(n)\) is defined by requiring the length of \(\alpha\) to be less than \(2k-1\).NEWLINENEWLINENEWLINEIt is clear that \(K'(n)\) implies \(K(n)\) and \(P(n)\) implies \(K'(n)\) and \(K(n)\). The main result of the paper under review (Theorem 3) states that every finitely generated group satisfying condition \(K(2)\) with respect to some finite generating set is finitely presentable. This implies that \(P(2)\)-groups are finitely presentable. In Theorem 4 the author proves that if \(G\) is a finitely generated group which satisfies condition \(K'(2)\) with respect to some finite generating set \(X\), then the following assertions hold: (a) The group \(G\) is finitely presentable and has solvable word problem. (b) The computational complexity of the word problem in \(G\) is at most exponential. That is, there is \(C>1\) and an algorithm solving the word problem in \(G\) with time-complexity at most \(C^n\) (where \(n\) is the length of the word in \(X\) being tested). By this theorem any \(P(2)\)-group has solvable word problem and hence it also has a recursive isoperimetric function.