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An obstruction to the strong relative hyperbolicity of a group. (English) Zbl 1188.20041
From the introduction: We give a simple combinatorial criterion for a group that, when satisfied, implies that the group cannot be strongly relatively hyperbolic. The criterion applies to several classes of groups, such as surface mapping class groups, Torelli groups, and automorphism and outer automorphism groups of free groups.
When a finitely generated group $$G$$ is strongly hyperbolic relative to a finite collection $$L_1,L_2,\dots,L_p$$ of proper subgroups, it is often possible to deduce that $$G$$ has a given property provided that the subgroups $$L_j$$ have the same property. In light of this, identifying a strong relatively hyperbolic group structure for a given group $$G$$, or indeed deciding whether or not one can exist, becomes an important objective.
The main result of this note, Theorem 2 in Section 3, asserts that such a structure cannot exist whenever the group $$G$$ satisfies a simple combinatorial property, namely that its commutativity graph with respect to some generating set $$S$$ is connected. We describe this graph in Section 3.
We then give elementary proofs that many groups of interest do not admit a strong relatively hyperbolic group structure. This is the case for all but finitely many surface mapping class groups, the Torelli group of a closed surface of genus at least 3, and the (special) automorphism and outer automorphism groups of almost all free groups. We note that this has already been established for many of these groups by other methods, and we indicate this where appropriate.

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 20F05 Generators, relations, and presentations of groups 57M07 Topological methods in group theory
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