On discrete hyperbolic arboreal groups.

*(English)*Zbl 0921.20044The author studies a certain class of groups generalizing free groups – the so-called discrete hyperbolic arboreal groups. The motivation comes from some difficult problems in model theory of free groups and, in particular, from the open question whether a nonabelian free group is positive-existentially closed in its profinite completion.

The paper is based on the notion of generalized tree which is an extension of the known notion of \(\Lambda\)-tree. For any linearly ordered abelian group \(\Lambda\) and any \(\Lambda\)-tree \((X,d)\), where \(d\) is a \(\Lambda\)-valued metric on \(X\), one can consider a symmetric ternary operation \(Y\), where \(Y(x_1,x_2,x_3)\) is defined to be the unique element \(x\in X\) such that \(d(x_i,x)+d(x,x_j)=d(x_i,x_j)\) for all distinct \(i\) and \(j\). The so defined operation \(Y\) satisfies the identities \(Y(x,y,x)=x\) and \(Y(Y(x,y,z),u,v)=Y(Y(x,u,v),y,Y(z,u,v))\). A generalized tree is defined to be a set \(X\) with a symmetric ternary operation \(Y\) on it satisfying the two identities. For \(a,b\in X\) denote the set \(\{Y(a,c,b):c\in X\}\) by \([a,b]\); such sets are called cells of the generalized tree. A generalized tree is said to be linear if for any \(a,b,c\in X\) with \(c\in[a,b]\) we have \([a,b]=[a,c]\cup[c,b]\). Note that the generalized trees arising from \(\Lambda\)-trees are linear.

An automorphism \(s\) of a linear generalized tree \(X\) is said to be hyperbolic if \(C\) and \(sC\) are inclusion-incomparable, for any cell \(C\) in \(X\). For a hyperbolic automorphism \(s\) let \(X_s\) denote the set \(\{x\in X:x\in[s^{-1}x,sx]\}\). A faithful action of a group \(G\) on a linear generalized tree \(X\) is said to be hyperbolic if any nonidentity element of \(G\) acts as a hyperbolic automorphism of \(X\) and, in addition, for any nonidentity \(s,t\in G\), if \(x,tx,tsx\in X_s\) then \(st=ts\). These conditions imply that \(G\) is torsion-free and the centralizers of nonidentity elements of \(G\) are abelian.

An arboreal group is defined to be a structure \((G,\cdot,Y)\), where \((G,\cdot)\) is a group, \((G,Y)\) is a linear generalized tree, and any left translation of the group is an automorphism of the generalized tree. The arboreal group is said to be hyperbolic if the action of the group on the generalized tree by left translations is hyperbolic. The author also introduces a notion of discrete hyperbolic action and the corresponding notion of discrete hyperbolic arboreal group.

In the paper under review he studies these notions in detail. One of the main results of the paper is that, given a discrete hyperbolic arboreal group \(G\) and a suitable family of abelian discrete hyperbolic arboreal groups which are convex extensions of maximal abelian arboreal subgroups of \(G\), the corresponding amalgamated sum has a canonical structure of discrete hyperbolic arboreal group. Some applications of this result to questions connected with the above-mentioned problem about free groups are given.

The paper is based on the notion of generalized tree which is an extension of the known notion of \(\Lambda\)-tree. For any linearly ordered abelian group \(\Lambda\) and any \(\Lambda\)-tree \((X,d)\), where \(d\) is a \(\Lambda\)-valued metric on \(X\), one can consider a symmetric ternary operation \(Y\), where \(Y(x_1,x_2,x_3)\) is defined to be the unique element \(x\in X\) such that \(d(x_i,x)+d(x,x_j)=d(x_i,x_j)\) for all distinct \(i\) and \(j\). The so defined operation \(Y\) satisfies the identities \(Y(x,y,x)=x\) and \(Y(Y(x,y,z),u,v)=Y(Y(x,u,v),y,Y(z,u,v))\). A generalized tree is defined to be a set \(X\) with a symmetric ternary operation \(Y\) on it satisfying the two identities. For \(a,b\in X\) denote the set \(\{Y(a,c,b):c\in X\}\) by \([a,b]\); such sets are called cells of the generalized tree. A generalized tree is said to be linear if for any \(a,b,c\in X\) with \(c\in[a,b]\) we have \([a,b]=[a,c]\cup[c,b]\). Note that the generalized trees arising from \(\Lambda\)-trees are linear.

An automorphism \(s\) of a linear generalized tree \(X\) is said to be hyperbolic if \(C\) and \(sC\) are inclusion-incomparable, for any cell \(C\) in \(X\). For a hyperbolic automorphism \(s\) let \(X_s\) denote the set \(\{x\in X:x\in[s^{-1}x,sx]\}\). A faithful action of a group \(G\) on a linear generalized tree \(X\) is said to be hyperbolic if any nonidentity element of \(G\) acts as a hyperbolic automorphism of \(X\) and, in addition, for any nonidentity \(s,t\in G\), if \(x,tx,tsx\in X_s\) then \(st=ts\). These conditions imply that \(G\) is torsion-free and the centralizers of nonidentity elements of \(G\) are abelian.

An arboreal group is defined to be a structure \((G,\cdot,Y)\), where \((G,\cdot)\) is a group, \((G,Y)\) is a linear generalized tree, and any left translation of the group is an automorphism of the generalized tree. The arboreal group is said to be hyperbolic if the action of the group on the generalized tree by left translations is hyperbolic. The author also introduces a notion of discrete hyperbolic action and the corresponding notion of discrete hyperbolic arboreal group.

In the paper under review he studies these notions in detail. One of the main results of the paper is that, given a discrete hyperbolic arboreal group \(G\) and a suitable family of abelian discrete hyperbolic arboreal groups which are convex extensions of maximal abelian arboreal subgroups of \(G\), the corresponding amalgamated sum has a canonical structure of discrete hyperbolic arboreal group. Some applications of this result to questions connected with the above-mentioned problem about free groups are given.

Reviewer: O.V.Belegradek (Kemerovo)

##### MSC:

20F65 | Geometric group theory |

20E05 | Free nonabelian groups |

03C60 | Model-theoretic algebra |

20A15 | Applications of logic to group theory |

20E18 | Limits, profinite groups |

20E08 | Groups acting on trees |

##### Keywords:

free groups; generalized trees; discrete hyperbolic arboreal groups; hyperbolic automorphisms; hyperbolic actions
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\textit{Ş. A. Basarab}, Commun. Algebra 26, No. 9, 2837--2865 (1998; Zbl 0921.20044)

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