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2 votes
0 answers
71 views

Empty preimage under homomorphism of finitely presented groups with decidable word problems

Let $G, H$ be finitely presented groups with decidable word problems. Can there be a homomorphism $f:G\to H$ such that there is no algorithm deciding given $w\in H$ whether $f^{-1}(w)$ is empty or not?...
user avatar
6 votes
2 answers
490 views

What is the name of this type of groups?

Suppose $A$ is a finite set and $\Sigma=A\cup A^{-1}$. Let $L\subseteq \Sigma^{\ast}$ be a regular language on the alphabet $\Sigma$. Is there a common name for the group $G$ presented as: $$G=\langle ...
Sh.M1972's user avatar
  • 2,233
6 votes
3 answers
872 views

An element $g$ in a group such that neither $g=1$ nor $g\ne 1$ can be proved.

Edited (this question contains two versions of a similar question) Is there some finitely presented group $G$ generated by $g_1,...,g_n$ such that there is an element $g\in G$ expressed as a finite ...
aglearner's user avatar
  • 14.3k
18 votes
1 answer
921 views

Automorphism group of the Turing degrees

It is conjectured that the automorphism group of the Turing degrees, $Aut(\mathcal{D})$, is trivial. However, to the best of my knowledge, the current state-of-the-art is that $Aut(\mathcal{D})$ is ...
Noah Schweber's user avatar
30 votes
3 answers
3k views

Is it decidable whether or not a collection of integer matrices generates a free group?

Suppose we have integer matrices $A_1,\ldots,A_n\in\operatorname{GL}(n,\mathbb Z)$. Define $\varphi:F_n\to\operatorname{GL}(n,\mathbb Z)$ by $x_i\mapsto A_i$. Is there an algorithm to decide whether ...
John Pardon's user avatar
  • 18.7k
17 votes
1 answer
875 views

Which finitely presented groups can be distinguished by decidable properties?

This question continues the line of inquiry of these three questions. Question. Which finitely presented groups can be distinguished by decidable properties? To be precise, let us say that φ is ...
Joel David Hamkins's user avatar
19 votes
4 answers
1k views

Does every decidable question about finitely presented groups amount to a question about abelian groups?

This question is about an issue left unresolved by Chad Groft's excellent question and John Stillwell's excellent answer of it. Since I find the possibility of an affirmative answer so tantalizing, I ...
Joel David Hamkins's user avatar
8 votes
1 answer
338 views

How bad can the recursive properties of finitely presented groups be?

Any finitely presented group naturally gives rise to an edge-labeled graph (the Cayley graph) and I am considering paths through this graph. Paths correspond to infinite sequences of generators, so ...
Aubrey da Cunha's user avatar