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On discrete hyperbolic arboreal groups. (English) Zbl 0921.20044
The 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.

##### 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
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##### References:
  Alperin R.C., Annals of Math, Studies 111 pp 265– (1987)  DOI: 10.1142/S0218196791000079 · Zbl 0722.20039 · doi:10.1142/S0218196791000079  DOI: 10.1016/0022-4049(91)90102-8 · Zbl 0741.20017 · doi:10.1016/0022-4049(91)90102-8  Basarab, S. 1992.The dual of the category of trees, Vol. 7, 21IMAR. Preprint  Basarab S., On a problem raised by Alperin and Bass II: Metric and order theoretic aspects 10 (1992)  Basarab S.A., Fundamenta Informaticae 30 (1997)  Bass H., Mathematical Sciences Research Institute Publications 19 pp 69– (1991)  Gaglione A.M., Houston Journal of Mathematics 19 (1993)  DOI: 10.1090/S0002-9947-1949-0032642-4 · doi:10.1090/S0002-9947-1949-0032642-4  DOI: 10.1090/S0002-9947-1972-0308283-0 · doi:10.1090/S0002-9947-1972-0308283-0  DOI: 10.1090/S0002-9947-1960-0151502-6 · doi:10.1090/S0002-9947-1960-0151502-6  Lyndon R.C., Combinatorial Group Theory (1977) · Zbl 0368.20023  Makanin G.S., Izvestia A.N. SSSR 48 pp 735– (1984)  DOI: 10.1112/blms/23.4.356 · Zbl 0754.20007 · doi:10.1112/blms/23.4.356  Razborov A.A., Izvestia A.N. SSSR 48 pp 779– (1984)  Remeslennikov V.N., Sibirskn Matem-aticeskii Jurnal 30 pp 193– (1989)  DOI: 10.1112/blms/25.1.37 · Zbl 0811.20026 · doi:10.1112/blms/25.1.37  Serre J.P., Trees (1980) · Zbl 0548.20018
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