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- Serres_property_FA abstract "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.".
- Serres_property_FA wikiPageID "2185021".
- Serres_property_FA wikiPageLength "3626".
- Serres_property_FA wikiPageOutDegree "22".
- Serres_property_FA wikiPageRevisionID "633982877".
- Serres_property_FA wikiPageWikiLink Algebraic_number_field.
- Serres_property_FA wikiPageWikiLink Amalgamated_product.
- Serres_property_FA wikiPageWikiLink Category:Properties_of_groups.
- Serres_property_FA wikiPageWikiLink Category:Trees_(graph_theory).
- Serres_property_FA wikiPageWikiLink Chevalley_group.
- Serres_property_FA wikiPageWikiLink Commutator_subgroup.
- Serres_property_FA wikiPageWikiLink Countable_set.
- Serres_property_FA wikiPageWikiLink Derived_group.
- Serres_property_FA wikiPageWikiLink Finitely_generated_group.
- Serres_property_FA wikiPageWikiLink Fixed_point_(mathematics).
- Serres_property_FA wikiPageWikiLink Free_product.
- Serres_property_FA wikiPageWikiLink Generating_set_of_a_group.
- Serres_property_FA wikiPageWikiLink Group_(mathematics).
- Serres_property_FA wikiPageWikiLink Group_action.
- Serres_property_FA wikiPageWikiLink Group_of_Lie_type.
- Serres_property_FA wikiPageWikiLink HNN_extension.
- Serres_property_FA wikiPageWikiLink Imaginary_quadratic_field.
- Serres_property_FA wikiPageWikiLink Index_of_a_subgroup.
- Serres_property_FA wikiPageWikiLink Jean-Pierre_Serre.
- Serres_property_FA wikiPageWikiLink Kazhdans_property_(T).
- Serres_property_FA wikiPageWikiLink Mathematics.
- Serres_property_FA wikiPageWikiLink Normal_subgroup.
- Serres_property_FA wikiPageWikiLink Quadratic_field.
- Serres_property_FA wikiPageWikiLink Quotient_group.
- Serres_property_FA wikiPageWikiLink Tree_(graph_theory).
- Serres_property_FA wikiPageWikiLinkText "Serre's property FA".
- Serres_property_FA hasPhotoCollection Serres_property_FA.
- Serres_property_FA wikiPageUsesTemplate Template:Cite_book.
- Serres_property_FA wikiPageUsesTemplate Template:Cite_journal.
- Serres_property_FA subject Category:Properties_of_groups.
- Serres_property_FA subject Category:Trees_(graph_theory).
- Serres_property_FA hypernym Property.
- Serres_property_FA type Building.
- Serres_property_FA 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.".
- Serres_property_FA label "Serre's property FA".
- Serres_property_FA sameAs m.06t7m2.
- Serres_property_FA sameAs Q7455404.
- Serres_property_FA sameAs Q7455404.
- Serres_property_FA wasDerivedFrom Serres_property_FAoldid=633982877.
- Serres_property_FA isPrimaryTopicOf Serres_property_FA.