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Results tagged with lo.logic
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user 1463
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
7
votes
Finite presentability and elementary equivalence
The following is not an answer to the main question, but provides some context which any answer may need to take into account. This context is certainly well known to many participants in the discussi …
- 25.2k
17
votes
Accepted
Is the diagonal of finitely presented groups computable?
The answer is "no". The hypotheses for the fibre product to be finitely presentable are given by the 1-2-3 theorem of Bridson--Baumslag--Miller--Short:
1-2-3 Theorem (BBMS): The fibre product $G\tim …
- 25.2k
3
votes
Accepted
What can the approximation of a group by some class be used for?
Let $\mathcal{F}_k$ be the class of free groups that can be generated by at most $k$ elements. In your terminology, limit groups are the groups that can be approximated by the groups in $\mathcal{F}_ …
- 25.2k
8
votes
Applications of logic to group theory?
In recent years, surely the most important and influential (and, unfortunately, sometimes controversial) work on the first-order theory of groups has been Sela's work on free and hyperbolic groups, an …
- 25.2k
6
votes
Accepted
Is there a non-Hopfian lacunary hyperbolic group?
Contrary to I think the expectations of the earlier answers, in a recent paper, Coulon--Guirardel showed that the answer to the title question is 'no'!
Theorem (Coulon--Guirardel): Every lacunary …
- 25.2k
21
votes
4
answers
2k
views
Is there a non-Hopfian lacunary hyperbolic group?
The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then …
- 25.2k
6
votes
Accepted
Do limit groups satisfy Howson's theorem?
The answer is 'yes'. A geometric proof, showing in fact that every finitely generated subgroup of a limit group is relatively quasiconvex, was given by Dahmani.
- 25.2k
8
votes
Decision problem on triviality of intersection of two subgroups
The OP asks for examples of groups with solvable word problem for which the intersection problem is undecidable. Such examples certainly exist. Here's one way to construct one.
Let $G$ be a finitely …
- 25.2k
14
votes
Is it decidable whether or not a collection of integer matrices generates a free group?
Here are some general facts that may be relevant.
Given a finitely presented group $G$ and a representation $\rho:G\to GL_n(\mathbb{Z})$, there is no algorithm which is uniform in $n$ that decides wh …
- 25.2k
5
votes
Does every decidable question about finitely presented groups amount to a question about ab...
One more counterexample. Makanin's algorithm solves systems of equations and inequations over any non-abelian free group F. The statement
'$G=\langle x_1,\ldots, x_m\mid r_1,\ldots,r_n\rangle$ sur …
- 25.2k
3
votes
Is any interesting question about a group G decidable from a presentation of G?
I don't have a complete answer, but here are some thoughts.
The Rips Construction takes an arbitrary finitely presented group Q and produces a 2-dimensional hyperbolic group $\Gamma$ and a short exa …
- 44.4k
1
vote
Is there a theorem that says that there is always more than one way to "continue a finite se...
Sonia, you need to define what you mean by "valid". Otherwise, I can continue in any way I want. For instance, here are two different continuations of the sequence "3,1,4,1":
3,1,4,1,0,0,0,0,0,0,0,0 …
- 25.2k