We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated to a finite Coxeter group form a latti...

There are many examples of non-isomorphic pairs of finitely generated abstract groups that are elementarily equivalent. We show that the situation in the category of profinite groups is different: If two finitely generated profinite groups are elementarily equivalent (as abstract groups), then they are isomorphic. The proof applies a result of Nikolov and Segal which in turn relies on the class...

 Let $R$ be a commutative ring with identity and $M$ be a finitely generated unital $R$-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module $M$ is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals o...

Let a connected real Lie group G act on a C real manifold M , and assume that the action is both C and effective on every nonempty G-invariant open set. We show that there is a nonempty G-invariant open subset U of some higher order frame bundle of M such that, for all u P U , the stabilizer in G of u is a discrete, finitely generated, free Abelian, central subgroup of G.

Funding: Acknowledgements. The authors would also like to thank the referee for their very careful and thorough reading of paper. This publication is in part a product visit first third author Mathematisches Forschungsinstitut Oberwolfach, Germany December 2016 as Research In Pairs program. was partially supported by NSF grants DMS–1600635 DMS-1854367.

Let n be a positive integer and FA`(n) be the free abelian latticeordered group on n generators. We prove that FA`(m) and FA`(n) do not satisfy the same first-order sentences in the language L={+,−, 0,∧,∨} if m 6= n. We also show that Th(FA`(n)) is decidable iff n ∈ {1, 2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices. A. M. S. C...

We show that the geometric part of the abelian étale fundamental group of a proper smooth variety over a local field is finitely generated over b Z with finite torsion, and describe its rank by the special fiber of the Néron model of the Albanese variety. As an application, we complete the class field theory of curves over local fields developed by S. Bloch and S. Saito, in which the theorem co...

We show that if Γ is a finitely presented metabelian group, then the “untwisted” fibre product or pull-back P associated to any short exact sequence 1 → N → Γ → Q → 1 is again finitely presented. In contrast, if N and Q are abelian, then the analogous “twisted” fibre-product is not finitely presented unless Γ is polycyclic. Also a number of examples are constructed, including a non-finitely pre...