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Statements

Subject Item
dbr:Serre's_property_FA
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Serre's property FA
rdfs:comment
In mathematics, Property FA is a property of groups first defined by Jean-Pierre Serre. A group G is said to have property FA if every action of G on a tree has a global fixed point. Serre shows that if a group has property FA, then it cannot split as an amalgamated product or HNN extension; indeed, if G is contained in an amalgamated product then it is contained in one of the factors. In particular, a finitely generated group with property FA has finite abelianization.
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dbc:Properties_of_groups dbc:Trees_(graph_theory)
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2185021
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888201353
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dbr:Derived_group dbr:Index_of_a_subgroup dbr:Chevalley_group dbr:Imaginary_quadratic_field dbr:Algebraic_number_field dbr:HNN_extension dbr:Tree_(graph_theory) dbc:Properties_of_groups dbr:Quotient_group dbr:Amalgamated_product dbr:Group_action_(mathematics) dbr:Group_(mathematics) dbr:Normal_subgroup dbr:Fixed_point_(mathematics) dbr:Kazhdan's_property_(T) dbr:Generating_set_of_a_group dbr:Countable_set dbc:Trees_(graph_theory) dbr:Mathematics dbr:Finitely_generated_group dbr:Jean-Pierre_Serre
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In mathematics, Property FA is a property of groups first defined by Jean-Pierre Serre. A group G is said to have property FA if every action of G on a tree has a global fixed point. Serre shows that if a group has property FA, then it cannot split as an amalgamated product or HNN extension; indeed, if G is contained in an amalgamated product then it is contained in one of the factors. In particular, a finitely generated group with property FA has finite abelianization. Property FA is equivalent for countable G to the three properties: G is not an amalgamated product; G does not have Z as a quotient group; G is finitely generated. For general groups G the third condition may be replaced by requiring that G not be the union of a strictly increasing sequence of subgroup. Examples of groups with property FA include SL3(Z) and more generally G(Z) where G is a simply-connected simple Chevalley group of rank at least 2. The group SL2(Z) is an exception, since it is isomorphic to the amalgamated product of the cyclic groups C4 and C6 along C2. Any quotient group of a group with property FA has property FA. If some subgroup of finite index in G has property FA then so does G, but the converse does not hold in general. If N is a normal subgroup of G and both N and G/N have property FA, then so does G. It is a theorem of Watatani that Kazhdan's property (T) implies property FA, but not conversely. Indeed, any subgroup of finite index in a T-group has property FA.
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dbr:Property
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