Preservation and distortion of area in finitely presented groups
The author develops the theory of distortion of combinatorial area in the Cayley graphs of finitely presented groups. Let \(G\) be a group with finite presentation \(\mathcal P\). The author defines an automorphism \(\phi\) of \(G\) to be \(\mathcal P\)-tame if \(\phi\) lifts to an automorphism of the free group on the generators of \(\mathcal P\) preserving the normal subgroup relations. He proves the following theorem: If \(\mathcal P\) is a finite one-relator presentation of \(G\) and if \(\phi\) is a \(\mathcal P\)-tame automorphism of \(G\), then there exists \(C>0\) so that for all loops \(w\) in the Cayley graph of \(\mathcal P\), we have \(\text{Area}_{\mathcal P}(w)\leq C \text{Area}_{\mathcal Q}(w)\), where \({\mathcal Q}\) is a finite presentation for the semi-direct product over \(\phi\) of \(G\) with the group \(\mathbb{Z}\) of integers. Here, the area \(\text{Area}_{\mathcal P}(w)\) of the loop \(w\) in the presentation \({\mathcal P}\) is the minimal number of 2-cells in a van Kampen diagram for \(w\). The author proves then that if \(K\) is the semi-direct product over \(\phi\) of the finitely presented group \(G\) with \(\mathbb{Z}\), then the area distortion of the subgroup \(G\) of \(K\) is at most an exponential of an isoperimetric function for \(K\). He presents examples of such a situation, where the inclusion \(G<K\) does not distort length but where area is distorted, and examples where length is distorted but area undistorted. The author introduces the exponent of area distortion \(\alpha(G,H)\) for the inclusion of a finitely presented group \(H\) in a finitely presented group \(G\), when \(H\) has a polynomial isoperimetric function of degree \(d\), and he proves that \(1\leq\alpha(G,H)\leq d\). Another result is the following suspension theorem: If \(\phi\in\text{GL}_n(\mathbb{Z})\) is of infinite order, then the exponent of area distortion of the torus-bundle over the circle with monodromy \(\phi\oplus 1\in\text{GL}_{n+1}(\mathbb{Z})\) is strictly greater than one. He gives then an example of an automatic subgroup \(H\) of an automatic group \(G\) where the exponent of area distortion is a number in the open interval \((1,2)\). He proves also that the inclusion of a finitely generated abelian group in the fundamental group of a compact 3-manifold does not distort area and he makes the relation with a result of D. Sullivan stating that if \(F\) is a transversely oriented codimension-1 foliation of a 3-manifold which has the property that through every leaf passes a transverse closed curve, then each leaf of \(F\) is quasi-area minimizing. The paper ends with a list of open problems.
Cayley graphs of finitely presented groups
scientific article; zbMATH DE number 912455
scientific article