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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.16885 |
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| _version_ | 1866911456716390400 |
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| author | Dudko, Artem Medynets, Constantine |
| author_facet | Dudko, Artem Medynets, Constantine |
| contents | Let $(X,T)$ be a Cantor minimal system, and let $Γ$ denote either its associated topological full group or the full group of a Bratteli diagram associated with $(X,T)$. In this paper we describe the structure of indecomposable (extreme) characters and the associated $\textrm{II}_1$-factor representations for the group $Γ$ and its commutator subgroup $Γ'$. In particular, we prove that: (1) for every nontrivial indecomposable character $χ$ of $Γ'$, there exists a finite collection (with repetitions allowed) $\{μ_i\}_{i\in I}$ of $T$-invariant ergodic measures on $X$ such that $χ(γ) = \prod_{i\in I} μ_i(Fix(γ))$, for every $γ\in Γ'$, where $Fix(γ) = \{x\in X : γx = x\}$; and (2) each indecomposable character of $Γ$ is the product of an indecomposable character of the form $\prod_{i\in I} μ_i(Fix(γ))$ and a homomorphism from $Γ$ into the unit circle.
As a consequence, we show that any finite-type unitary representation of $Γ'$ that does not contain a regular subrepresentation is automatically continuous with respect to the uniform topology on $Γ'$.
We also establish a general result on automatic continuity of finite-type unitary representations of infinite groups, which we use in our proofs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16885 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characters and $II_1$-Factor Representations of Full Groups of Cantor Minimal Systems Dudko, Artem Medynets, Constantine Group Theory Primary 37B05, 22D25, 22C32, Secondary 46L36, 37A20, 37A05, 54H15 Let $(X,T)$ be a Cantor minimal system, and let $Γ$ denote either its associated topological full group or the full group of a Bratteli diagram associated with $(X,T)$. In this paper we describe the structure of indecomposable (extreme) characters and the associated $\textrm{II}_1$-factor representations for the group $Γ$ and its commutator subgroup $Γ'$. In particular, we prove that: (1) for every nontrivial indecomposable character $χ$ of $Γ'$, there exists a finite collection (with repetitions allowed) $\{μ_i\}_{i\in I}$ of $T$-invariant ergodic measures on $X$ such that $χ(γ) = \prod_{i\in I} μ_i(Fix(γ))$, for every $γ\in Γ'$, where $Fix(γ) = \{x\in X : γx = x\}$; and (2) each indecomposable character of $Γ$ is the product of an indecomposable character of the form $\prod_{i\in I} μ_i(Fix(γ))$ and a homomorphism from $Γ$ into the unit circle. As a consequence, we show that any finite-type unitary representation of $Γ'$ that does not contain a regular subrepresentation is automatically continuous with respect to the uniform topology on $Γ'$. We also establish a general result on automatic continuity of finite-type unitary representations of infinite groups, which we use in our proofs. |
| title | Characters and $II_1$-Factor Representations of Full Groups of Cantor Minimal Systems |
| topic | Group Theory Primary 37B05, 22D25, 22C32, Secondary 46L36, 37A20, 37A05, 54H15 |
| url | https://arxiv.org/abs/2602.16885 |