1,529 research outputs found
The Hyperbolic Ax-Lindemann-Weierstrass conjecture
The hyperbolic Ax-Lindemann-Weierstrass conjecture is a functional algebraic
independence statement for the uniformizing map of an arithmetic variety. In
this paper we provide a proof of this conjecture, generalizing previous work of
Pila-Tsimerman and Peterzil-Starchenko.Comment: The only modification is A.Yafaev'acknowledgement of ERC suppor
Two remarks on elementary theories of groups obtained by free constructions
We give two slight generalizations of results of Poizat about elementary
theories of groups obtained by free constructions. The first-one concerns the
non-superstability of such groups in most cases, and the second-one concerns
the connectedness of most free products of groups
What does a group algebra of a free group know about the group?
We describe solutions to the problem of elementary classification in the
class of group algebras of free groups. We will show that unlike free groups,
two group algebras of free groups over infinite fields are elementarily
equivalent if and only if the groups are isomorphic and the fields are
equivalent in the weak second order logic. We will show that the set of all
free bases of a free group is 0-definable in the group algebra when
is an infinite field, the set of geodesics is definable, and many geometric
properties of are definable in . Therefore knows some very
important information about . We will show that similar results hold for
group algebras of limit groups.Comment: Published, Available for free at
https://www.sciencedirect.com/science/article/pii/S0168007218300174?dgcid=STMJ_73515_AUTH_SERV_PPUB_V38
arXiv admin note: text overlap with arXiv:1509.0411
The definability criterions for convex projective polyhedral reflection groups
Following Vinberg, we find the criterions for a subgroup generated by
reflections \Gamma \subset \SL^{\pm}(n+1,\mathbb{R}) and its finite-index
subgroups to be definable over where is an integrally
closed Noetherian ring in the field . We apply the criterions for
groups generated by reflections that act cocompactly on irreducible properly
convex open subdomains of the -dimensional projective sphere. This gives a
method for constructing injective group homomorphisms from such Coxeter groups
to \SL^{\pm}(n+1,\mathbb{Z}). Finally we provide some examples of
\SL^{\pm}(n+1,\mathbb{Z})-representations of such Coxeter groups. In
particular, we consider simplicial reflection groups that are isomorphic to
hyperbolic simplicial groups and classify all the conjugacy classes of the
reflection subgroups in \SL^{\pm}(n+1,\mathbb{R}) that are definable over
. These were known by Goldman, Benoist, and so on previously.Comment: 31 pages, 8 figure
On ampleness and pseudo-Anosov homeomorphisms in the free group
We use pseudo-Anosov homeomorphisms of surfaces in order to prove that the
first order theory of non abelian free groups, , is -ample for any
. This result adds to the work of Pillay, that proved that
is non CM -trivial. The sequence witnessing ampleness is a sequence of
primitive elements in .
Our result provides an alternative proof to the main result of a preprint by
Ould Houcine-Tent.
We also add an appendix in which we make a few remarks on Sela's paper on
imaginaries in torsion free hyperbolic groups. In particular we give
alternative transparent proofs concerning the non-elimination of certain
imaginaries.Comment: 22 pages, 2 figures. To appear in the Turkish Journal of Mathematics.
Replaces arXiv:1205.466
Forking and JSJ decompositions in the free group II
We give a complete characterization of the forking independence relation over
any set of parameters in the free groups of finite rank, in terms of the
decompositions relative to those parameters.Comment: 23 pages. In the updated version, the (cumbersome) generalization of
the main result given in Theorem 3.15 of the first version was removed, as
well as its proof. This induced some substantial simplifications in the
presentation. A number of examples and figures have also been added to help
the reade
Fields definable in the free group
We prove that no infinite field is definable in the theory of the free groupComment: First version, 47 pages, 7 figure
The Riemann Mapping Theorem for semianalytic domains and o-minimality
We consider the Riemann Mapping Theorem in the case of a bounded simply
connected and semianalytic domain. We show that the germ at 0 of the Riemann
map (i.e. biholomorphic map) from the upper half plane to such a domain can be
realized in a certain quasianalytic class if the angle of the boundary at the
point to which 0 is mapped, is greater than 0. This quasianalytic class was
introduced and used by Ilyashenko in his work on Hilbert's 16th problem. With
this result we can prove that the Riemann map from a bounded simply connected
semianalytic domain onto the unit ball is definable in an o-minimal structure,
provided that at singular boundary points the angles of the boundary are
irrational multiples of .Comment: 22 page
The free group does not have the finite cover property
We prove that the first order theory of nonabelian free groups eliminates the
"there exists infinitely many" quantifier (in eq). Equivalently, since the
theory of nonabelian free groups is stable, it does not have the finite cover
property. We also extend our results to torsion-free hyperbolic groups under
some conditions.Comment: 23 pages, to appear in the Israel J. Mat
Equations in Algebras
We show that the Diophantine problem(decidability of equations) is
undecidable in free associative algebras over any field and in the group
algebras over any field of a wide variety of torsion free groups, including
toral relatively hyperbolic groups, right angled Artin groups, commutative
transitive groups, the fundamental groups of various graph groups, etc
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