Tarski’s theorem on amenable groups

Target entity: Tarski’s theorem on amenable groups
Context triple: [Banach–Tarski paradox, relatedTo, Tarski’s theorem on amenable groups]

• A. Kesten’s theorem on random walks on groups
  Kesten’s theorem on random walks on groups is a fundamental result in probability theory that characterizes amenability of groups via the spectral radius of associated random walks.
• B. Connes embedding problem
  The Connes embedding problem is a central open question in operator algebras and quantum theory that asks whether every separable II₁ factor can be approximated in a specific way by finite-dimensional matrix algebras.
• C. Dehn’s decision problems in group theory
  Dehn’s decision problems in group theory are foundational early 20th-century problems that introduced algorithmic questions about the solvability of word, conjugacy, and isomorphism problems in finitely presented groups, helping launch the field of algorithmic group theory.
• D. Dixmier problem in group theory
  The Dixmier problem in group theory is a famous open question asking whether every nontrivial finitely generated group has a nontrivial finite-dimensional unitary representation.
• E. Gowers dichotomy for Banach spaces
  Gowers dichotomy for Banach spaces is a fundamental result in functional analysis that classifies infinite-dimensional Banach spaces by showing that each contains either a subspace with an unconditional basis or a hereditarily indecomposable subspace.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Tarski’s theorem on amenable groups
Target entity description: Tarski’s theorem on amenable groups is a fundamental result in group theory and measure theory that characterizes amenable groups as precisely those that do not admit Banach–Tarski-type paradoxical decompositions.

• A. Kesten’s theorem on random walks on groups
  Kesten’s theorem on random walks on groups is a fundamental result in probability theory that characterizes amenability of groups via the spectral radius of associated random walks.
• B. Connes embedding problem
  The Connes embedding problem is a central open question in operator algebras and quantum theory that asks whether every separable II₁ factor can be approximated in a specific way by finite-dimensional matrix algebras.
• C. Dehn’s decision problems in group theory
  Dehn’s decision problems in group theory are foundational early 20th-century problems that introduced algorithmic questions about the solvability of word, conjugacy, and isomorphism problems in finitely presented groups, helping launch the field of algorithmic group theory.
• D. Dixmier problem in group theory
  The Dixmier problem in group theory is a famous open question asking whether every nontrivial finitely generated group has a nontrivial finite-dimensional unitary representation.
• E. Gowers dichotomy for Banach spaces
  Gowers dichotomy for Banach spaces is a fundamental result in functional analysis that classifies infinite-dimensional Banach spaces by showing that each contains either a subspace with an unconditional basis or a hereditarily indecomposable subspace.
• F. None of above. chosen