Characterizing Lie groups by controlling their zero-dimensional subgroups
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Dikran Dikranjan
Abstract
We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The “compact-like” properties we consider include (local) compactness, (local) ω-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is A sample of our characterizations is as follows:
(i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups.
(ii) An abelian topological group G is a Lie group if and only if G is locally minimal, locally precompact and all closed metric zero-dimensional subgroups of G are discrete.
(iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups.
(iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 22540089
Award Identifier / Grant number: L13710
Funding statement: The first named author gratefully acknowledges a JSPS long term visitor grant L13710, as well as the generous hospitality of the Division of Mathematics, Physics and Earth Sciences at the Graduate School of Science and Engineering of Ehime University, Matsuyama. The second named author was partially supported by the Grant-in-Aid for Scientific Research (C) No. 22540089 by the Japan Society for the Promotion of Science (JSPS).
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Articles in the same Issue
- Frontmatter
- The variance of divisor sums in arithmetic progressions
- Characterizing Lie groups by controlling their zero-dimensional subgroups
- CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
- Towards a Goldberg–Shahidi pairing for classical groups
- On the non-existence of cyclic splitting fields for division algebras
- Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
- Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
- Anharmonic solutions to the Riccati equation and elliptic modular functions
- A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
- Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
- On symplectic semifield spreads of PG(5,q2), q odd
- From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
- Golodness and polyhedral products for two-dimensional simplicial complexes
Articles in the same Issue
- Frontmatter
- The variance of divisor sums in arithmetic progressions
- Characterizing Lie groups by controlling their zero-dimensional subgroups
- CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
- Towards a Goldberg–Shahidi pairing for classical groups
- On the non-existence of cyclic splitting fields for division algebras
- Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
- Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
- Anharmonic solutions to the Riccati equation and elliptic modular functions
- A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
- Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
- On symplectic semifield spreads of PG(5,q2), q odd
- From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
- Golodness and polyhedral products for two-dimensional simplicial complexes
