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Abstract Let be a commutative ring, and assume that every non‐trivial ideal of has finite index. We show that if has bounded elementary generation then every conjugation‐invariant norm on it is either discrete or precompact. If is any group satisfying this dichotomy, we say that has the dichotomy property . We relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the celebrated normal subgroup theorem of Margulis and the seminal work of Nikolov and Segal. As a consequence we derive constraints to the possible approximations of certain non‐residually finite central extensions of arithmetic groups, which we hope might have further applications in the study of sofic groups. In the last section we provide several open problems for further research.
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Words, Hausdorff dimension and randomly free groups
We bound the size of fibers of word maps in finite and residually finite groups, and derive various applications. Our main result shows that, for any word $$1 \ne w \in F_d$$ there exists $$\e > 0$$ such that if $$\Gamma$$ is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map $$w:\Gamma^d \rightarrow \Gamma$$ have Hausdorff dimension at most $$d -\e$$. We conclude that profinite groups $$G := \hat\Gamma$$, $$\Gamma$$ as above, satisfy no probabilistic identity, and therefore they are \emph{randomly free}, namely, for any $$d \ge 1$$, the probability that randomly chosen elements $$g_1, \ldots , g_d \in G$$ freely generate a free subgroup (isomorphic to $$F_d$$) is $$1$$. This solves an open problem from \cite{DPSS}. Additional applications and related results are also established. For example, combining our results with recent results of Bors, we conclude that a profinite group in which the set of elements of finite odd order has positive measure has an open prosolvable subgroup. This may be regarded as a probabilistic version of the Feit-Thompson theorem.
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- Award ID(s):
- 1702152
- PAR ID:
- 10095895
- Date Published:
- Journal Name:
- Mathematische Annalen
- Volume:
- 371
- Issue:
- 3-4
- ISSN:
- 0025-5831
- Page Range / eLocation ID:
- 1409-1427
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/https://doi.org/10.1007/s00208-017-1635-y
