##
**Tree-graded spaces and asymptotic cones of groups. (With an appendix by Denis Osin and Mark Sapir).**
*(English)*
Zbl 1101.20025

The authors present solutions for some problems in asymptotic geometry of relatively hyperbolic groups. The main tool is the concept of tree-graded and asymptotically tree-graded spaces. A complete geodesic metric space \(F\) is called ‘tree-graded’ with respect to a collection \(\mathcal P\) of closed geodesic subsets (called ‘pieces’), if the following two properties are satisfied:

(\(T_1\)) every two different pieces have at most one common point;

(\(T_2\)) every simple geodesic triangle (a single point is considered as a degenerate simple triangle) in \(F\) is contained in a piece.

The space \(F\) is ‘asymptotically tree-graded’ with respect to the family \({\mathcal A}=\{A_i\subset F\mid i\in I\}\) if every asymptotic cone is tree-graded with respect to the family \(\mathcal A_\omega\) of ultralimits \(\lim^\omega(A_{i_n})\), where \(\omega\) is a non-principal ultrafilter.

The main interest in the notion of asymptotically tree-graded space resides in the following characterization of relatively hyperbolic groups. A finitely generated group \(G\) is hyperbolic relative to finitely generated subgroups \(H_1,\dots,H_n\) if and only if it is asymptotically tree-graded with respect to the family \({\mathcal H}=\{H_1,\dots,H_n\}\). The “if” statement is proven in the Appendix.

The main results of the paper are contained in several theorems. It is shown that the class of groups that are hyperbolic relative to unconstricted subgroups is closed under quasi-isometry. A finitely generated group is called ‘unconstricted’ if one of its asymptotic cones has no global cut-points. Let \(G\) be a finitely generated group that is hyperbolic relative to unconstricted subgroups \(H_1,\dots,H_m\) and \(G'\) be a group that is quasi-isometric to \(G\). Then \(G'\) is hyperbolic relative to subgroups \(H'_1,\dots,H'_m\), each of which is quasi-isometric to one of \(H_1,\dots,H_m\).

Another theorem gives additional information about the possible structure of fundamental groups of asymptotic cones. For every countable group \(C\) the free product of continuously many copies of \(C\) is the fundamental group of an asymptotic cone of a 2-generated group, and there exists a 2-generated group \(\Gamma\) such that for every finitely presented group \(G\), the free product of continuously many copies of \(G\) is the fundamental group of an asymptotic cone of \(\Gamma\).

The third theorem answers a question of M. Gromov [Geometric group theory, Vol. 2: Asymptotic invariants of infinite groups. Lond. Math. Soc. Lect. Note Ser. 182 (1993; Zbl 0841.20039)] about the number of asymptotic cones of a finitely generated group. Regardless of whether the Continuum Hypothesis is true or not, there exists a finitely generated group \(G\) with continuously many pairwise non-\(\pi_1\)-equivalent asymptotic cones. Note that by a result of L. Kramer, S. Shelah, K. Tent and S. Thomas [Adv. Math. 193, No. 1, 142-173 (2005; Zbl 1139.22010)], continuum is the maximal possible number of different asymptotic cones of a finitely generated group, provided that the Continuum Hypothesis is true.

As an application of the tree-graded spaces concept, some geometric properties of the Cayley graph \(\text{Cayley}(G,S)\) of a relatively hyperbolic group \(G=\langle S\rangle\) are found. The most remarkable is the Morse property: for a group \(G\) hyperbolic relative to the collection of subgroups \(H_1,\dots,H_m\) there exists a constant \(M\) depending only on the generating set \(S\) such that for every \(L\geq 1\), \(C \geq 0\) there exists \(\tau:=\tau(L,C,S)\), such that each \((L,C)\)-quasi-geodesic \(q\) in \(\text{Cayley}(G,S)\) is contained in the \(\tau\)-tubular neighborhood of the \(M\)-saturation of any geodesic \(g\) with the same endpoints. Here ‘\(M\)-saturation’ of the geodesic \(g\) in \(\text{Cayley}(G,S)\) is the union of \(g\) and all left cosets of \(H_i\) whose \(M\)-tubular neighborhoods intersect \(g\). A list of open problems is included.

(\(T_1\)) every two different pieces have at most one common point;

(\(T_2\)) every simple geodesic triangle (a single point is considered as a degenerate simple triangle) in \(F\) is contained in a piece.

The space \(F\) is ‘asymptotically tree-graded’ with respect to the family \({\mathcal A}=\{A_i\subset F\mid i\in I\}\) if every asymptotic cone is tree-graded with respect to the family \(\mathcal A_\omega\) of ultralimits \(\lim^\omega(A_{i_n})\), where \(\omega\) is a non-principal ultrafilter.

The main interest in the notion of asymptotically tree-graded space resides in the following characterization of relatively hyperbolic groups. A finitely generated group \(G\) is hyperbolic relative to finitely generated subgroups \(H_1,\dots,H_n\) if and only if it is asymptotically tree-graded with respect to the family \({\mathcal H}=\{H_1,\dots,H_n\}\). The “if” statement is proven in the Appendix.

The main results of the paper are contained in several theorems. It is shown that the class of groups that are hyperbolic relative to unconstricted subgroups is closed under quasi-isometry. A finitely generated group is called ‘unconstricted’ if one of its asymptotic cones has no global cut-points. Let \(G\) be a finitely generated group that is hyperbolic relative to unconstricted subgroups \(H_1,\dots,H_m\) and \(G'\) be a group that is quasi-isometric to \(G\). Then \(G'\) is hyperbolic relative to subgroups \(H'_1,\dots,H'_m\), each of which is quasi-isometric to one of \(H_1,\dots,H_m\).

Another theorem gives additional information about the possible structure of fundamental groups of asymptotic cones. For every countable group \(C\) the free product of continuously many copies of \(C\) is the fundamental group of an asymptotic cone of a 2-generated group, and there exists a 2-generated group \(\Gamma\) such that for every finitely presented group \(G\), the free product of continuously many copies of \(G\) is the fundamental group of an asymptotic cone of \(\Gamma\).

The third theorem answers a question of M. Gromov [Geometric group theory, Vol. 2: Asymptotic invariants of infinite groups. Lond. Math. Soc. Lect. Note Ser. 182 (1993; Zbl 0841.20039)] about the number of asymptotic cones of a finitely generated group. Regardless of whether the Continuum Hypothesis is true or not, there exists a finitely generated group \(G\) with continuously many pairwise non-\(\pi_1\)-equivalent asymptotic cones. Note that by a result of L. Kramer, S. Shelah, K. Tent and S. Thomas [Adv. Math. 193, No. 1, 142-173 (2005; Zbl 1139.22010)], continuum is the maximal possible number of different asymptotic cones of a finitely generated group, provided that the Continuum Hypothesis is true.

As an application of the tree-graded spaces concept, some geometric properties of the Cayley graph \(\text{Cayley}(G,S)\) of a relatively hyperbolic group \(G=\langle S\rangle\) are found. The most remarkable is the Morse property: for a group \(G\) hyperbolic relative to the collection of subgroups \(H_1,\dots,H_m\) there exists a constant \(M\) depending only on the generating set \(S\) such that for every \(L\geq 1\), \(C \geq 0\) there exists \(\tau:=\tau(L,C,S)\), such that each \((L,C)\)-quasi-geodesic \(q\) in \(\text{Cayley}(G,S)\) is contained in the \(\tau\)-tubular neighborhood of the \(M\)-saturation of any geodesic \(g\) with the same endpoints. Here ‘\(M\)-saturation’ of the geodesic \(g\) in \(\text{Cayley}(G,S)\) is the union of \(g\) and all left cosets of \(H_i\) whose \(M\)-tubular neighborhoods intersect \(g\). A list of open problems is included.

Reviewer: P. D. Andreev (Arkhangelsk)

### MathOverflow Questions:

Proof that lifts of geodesics are quasi-geodesics (relatively hyperbolic groups)### MSC:

20F67 | Hyperbolic groups and nonpositively curved groups |

20F65 | Geometric group theory |

20F69 | Asymptotic properties of groups |

20F05 | Generators, relations, and presentations of groups |

57M07 | Topological methods in group theory |