On quasiconvex null sequences admit quasiconvex null sequences? - PowerPoint PPT Presentation

TOPOSYM 2016 Außenhofer Motivation Quasi–convex sets Which groups On quasi–convex null sequences admit quasi–convex null sequences? The main result Lydia Außenhofer Open questions lydia.aussenhofer@uni-passau.de Bibliography TOPOSYM 2016

TOPOSYM 2016 Außenhofer Motivation Quasi–convex sets Which groups admit quasi–convex null sequences? On quasi–convex null sequences The main result Open questions Bibliography

TOPOSYM 2016 Außenhofer Motivation 1 Motivation Quasi–convex sets Quasi–convex sets 2 Which groups admit quasi–convex null sequences? Which groups admit quasi–convex null sequences? 3 The main result Open The main result 4 questions Bibliography 5 Open questions Bibliography 6

TOPOSYM 2016 Außenhofer Motivation Quasi–convex sets Notation Which groups admit quasi–convex A subset C of a real vector space V is called convex if null sequences? λ x +( 1 − λ ) y ∈ C ∀ λ ∈ [ 0 , 1 ] , ∀ x , y ∈ C . The main result Open questions Bibliography In particular, if x , y ∈ C are two different points, then { λ x +( 1 − λ ) y : λ ∈ [ 0 , 1 ] } ⊆ C and hence | C | ≥ c .

TOPOSYM 2016 Außenhofer In order to define ” convex ” sets in an abelian topological group, we use the description of closed, symmetric convex Motivation subsets of real locally convex vector spaces given by the Quasi–convex sets Hahn Banach theorem: Which groups admit quasi–convex null Theorem sequences? The main Let V be a real locally convex vector space and 0 ∈ C ⊆ V. result Then the following assertion are equivalent: Open questions C is closed, symmetric and convex. 1 Bibliography For every x / ∈ C there exists a continuous linear form 2 f : V → R such that � � − 1 4 , 1 | f ( x ) | > 1 f ( C ) ⊆ and 4 . 4

TOPOSYM 2016 Außenhofer Motivation Notation Quasi–convex sets Which groups T + = [ − 1 4 , 1 The torus T := R / Z , 4 ]+ Z admit quasi–convex null Definition sequences? The main Let ( G , τ ) be an abelian topological group. result Open questions G ∧ := ( G , τ ) ∧ := { χ : G → T | χ is a continuous hom. } Bibliography is under pointwise addition an abelian group. It is called dual group or character group .

TOPOSYM 2016 Außenhofer Quasi–convex sets Motivation Quasi–convex sets Definition Which groups For a subset A of a topological group ( G , ρ ) we define the admit quasi–convex polar of A as null sequences? A ⊲ := { χ ∈ G ∧ | ∀ x ∈ A χ ( x ) ∈ T + } The main result Open and for B ⊆ G ∧ we define the pre–polar of B by questions Bibliography B ⊳ := { x ∈ G | ∀ χ ∈ B χ ( x ) ∈ T + } . A subset A of a topological group ( G , τ ) is called quasi–convex if for every x ∈ G \ A there exists a character χ ∈ A ⊲ such that χ ( x ) / ∈ T + .

TOPOSYM 2016 Außenhofer Motivation Quasi–convex sets Which groups Locally quasi–convex groups admit quasi–convex null sequences? Definition (Vilenkin; 1951) The main result A topological group ( G , τ ) is called locally quasi–convex if Open questions it has a neighborhood basis at 0 consisting of quasi–convex Bibliography sets.

TOPOSYM 2016 Examples of quasi–convex sets Außenhofer Motivation Example Quasi–convex sets T + ⊆ T is quasi–convex. 1 Which groups admit � � − 1 4 m , 1 T m := + Z ⊆ T is quasi–convex. quasi–convex 2 4 m null sequences? The intersection of quasi–convex sets is quasi–convex. 3 The main The inverse image of a quasi–convex set under a result 4 Open continuous homomorphism is quasi–convex. questions For B ⊆ G ∧ the set 5 Bibliography � χ − 1 ( T m ) ( B , T m ) := χ ∈ B is quasi–convex. For every A ⊆ G the set ( A ⊲ ) ⊳ = ( A ⊲ , T + ) is 6 quasi–convex.

TOPOSYM 2016 Außenhofer Motivation The quasi–convex hull Quasi–convex sets Which groups Proposition admit quasi–convex For a subset A of an abelian topological group ( G , τ ) the set null sequences? The main ( A ⊲ ) ⊳ result Open questions is the smallest quasi–convex set containing A. It is called Bibliography the quasi–convex hull of A and denoted by qc ( A ) . Corollary A ⊆ G is quasi–convex iff A = ( A ⊲ ) ⊳ .

TOPOSYM 2016 Außenhofer Motivation Quasi–convex Examples of locally quasi–convex groups sets Which groups admit quasi–convex Example null sequences? A Hausdorff topological vector space is locally convex 1 The main result iff it is locally quasi–convex. Open Every character group endowed with the questions 2 compact–open topology is locally quasi–convex. Bibliography Every locally compact abelian (LCA for short) group is 3 locally quasi–convex.

TOPOSYM 2016 Cardinality of quasi–convex sets Außenhofer Motivation Proposition Quasi–convex sets Every symmetric, closed and convex subset of a locally Which groups convex vector space is locally quasi–convex. Hence there admit quasi–convex exist quasi–convex sets of cardinality ≥ c . null sequences? The main result Proposition (L.A. 1998) Open If G is an MAP group, then questions Bibliography qc ( x ) = { x , − x , 0 } for every x ∈ G. Proposition (L.A. 1998; Dikranjan, Kunen 2007) If G is an MAP group, then the quasi–convex hull of every finite subset is finite.

TOPOSYM 2016 Außenhofer Motivation Quasi–convex sets Which groups admit quasi–convex Question null sequences? The main Question result Open Are there countably infinite quasi–convex sets? questions Bibliography

TOPOSYM 2016 Quasi–convex null sequences Außenhofer Motivation Definition Quasi–convex A sequence ( x n ) n ∈ N in an abelian topological group is called sets a quasi–convex null sequence if Which groups admit quasi–convex null x n → 0 sequences? The main result and the set Open questions { x n : n ∈ N }∪{− x n : n ∈ N }∪{ 0 } Bibliography is quasi–convex. Question Are there quasi–convex null sequences? 1 Which (LCA) groups have quasi–convex null 2 sequences?

TOPOSYM 2016 Außenhofer Motivation Quasi–convex sets Quasi–convex null sequences in T Which groups admit quasi–convex null Theorem (L. de Leo 2008) sequences? The main Let ( a n ) ∈ N N with a n + 1 − a n ≥ 2 for all n ∈ N . Then result Open ( 2 − a n + 1 + Z ) n ∈ N questions Bibliography is a quasi-convex null sequence in T .

TOPOSYM Quasi–convex null sequences in R and J 2 2016 Außenhofer Theorem (D. Dikranjan, L. de Leo, 2010) Motivation Let ( a n ) ∈ N N with a n + 1 − a n ≥ 2 for all n ∈ N . Then Quasi–convex sets Which groups ( 2 − a n + 1 ) n ∈ N admit quasi–convex null sequences? is a quasi-convex null sequence in R . The main result Example Open questions qc ( { 2 − n : n ∈ N 0 } ) = [ − 1 , 1 ] ⊆ R Bibliography Theorem (D. Dikranjan, L. de Leo, 2010) Let ( a n ) ∈ N N with a n + 1 − a n ≥ 2 for all n ∈ N . Then ( 2 a n − 1 ) n ∈ N is a quasi-convex null sequence in J 2 .

TOPOSYM 2016 Außenhofer Quasi–convex sets in bounded groups Motivation Quasi–convex Proposition (D. Dikranjan, G. Luk´ acs; 2010) sets Which groups Every abelian topological group of exponent ≤ 3 has no admit quasi–convex non–trivial quasi–convex null sequence. null sequences? The main Proof. result Open Let G be a bounded abelian topological group of exponent questions ≤ 3. Fix x ∈ G and χ ∈ G ∧ .If χ ( x ) � = 0 + Z , then χ ( x ) / ∈ T + . Bibliography This implies { x } ⊲ = { x } ⊥ . Hence, if ( x n ) is a null sequence, then { x n : n ∈ N } ⊲ is a subgroup of G ∧ and so is qc ( { x n : n ∈ N } ) = ( { x n : n ∈ N } ⊥ ) ⊳ . In particular, qc ( { x n : n ∈ N } ) is a subgroup of G , hence a homogeneous space. This yields { 0 }∪{± x n : n ∈ N } � = qc ( { x n : n ∈ N } ) .

TOPOSYM 2016 Außenhofer Motivation Quasi–convex sets Which groups The groups Z κ 2 and Z κ admit 3 quasi–convex null sequences? The main Corollary (D. Dikranjan, G. Luk´ acs; 2010) result The groups Z κ 2 and Z κ Open 3 do not admit a non–trivial questions quasi–convex null sequence. Bibliography

TOPOSYM 2016 Außenhofer Motivation Quasi–convex Quasi–convex null sequences in LCA groups sets Which groups admit quasi–convex Theorem (D. Dikranjan, G. Luk´ acs; 2010) null sequences? For a LCA group G the following assertions are equivalent: The main result G has no non–trivial quasi–convex null sequence. 1 Open Either the subgroup G [ 2 ] = { x ∈ G : 2 x = 0 } or questions 2 Bibliography G [ 3 ] = { x ∈ G : 3 x = 0 } is open in G. G contains a compact open subgroup topologically 3 isomorphic to Z κ 2 or Z κ 3 ( κ a cardinal).

TOPOSYM 2016 Außenhofer Motivation Quasi–convex sets Which groups admit Quasi–convex null sequences in LCA groups quasi–convex null sequences? The main Question result Do similar results hold for arbitrary precompact abelian Open questions groups? Bibliography