699 research outputs found
Algorithmic recognition of infinite cyclic extensions
We prove that one cannot algorithmically decide whether a finitely presented
-extension admits a finitely generated base group, and we use this
fact to prove the undecidability of the BNS invariant. Furthermore, we show the
equivalence between the isomorphism problem within the subclass of unique
-extensions, and the semi-conjugacy problem for deranged outer
automorphisms.Comment: 24 page
Unification modulo a partial theory of exponentiation
Modular exponentiation is a common mathematical operation in modern
cryptography. This, along with modular multiplication at the base and exponent
levels (to different moduli) plays an important role in a large number of key
agreement protocols. In our earlier work, we gave many decidability as well as
undecidability results for multiple equational theories, involving various
properties of modular exponentiation. Here, we consider a partial subtheory
focussing only on exponentiation and multiplication operators. Two main results
are proved. The first result is positive, namely, that the unification problem
for the above theory (in which no additional property is assumed of the
multiplication operators) is decidable. The second result is negative: if we
assume that the two multiplication operators belong to two different abelian
groups, then the unification problem becomes undecidable.Comment: In Proceedings UNIF 2010, arXiv:1012.455
Max Dehn, Axel Thue, and the Undecidable
This is a short essay on the roles of Max Dehn and Axel Thue in the
formulation of the word problem for (semi)groups, and the story of the proofs
showing that the word problem is undecidable.Comment: Definition of undecidability and unsolvability improve
Orbit decidability and the conjugacy problem for some extensions of groups
Given a short exact sequence of groups with certain conditions, , we prove that has solvable conjugacy problem if and only if
the corresponding action subgroup is orbit decidable. From
this, we deduce that the conjugacy problem is solvable, among others, for all
groups of the form , , , and with virtually solvable action
group . Also, we give an easy way of constructing
groups of the form and with
unsolvable conjugacy problem. On the way, we solve the twisted conjugacy
problem for virtually surface and virtually polycyclic groups, and give an
example of a group with solvable conjugacy problem but unsolvable twisted
conjugacy problem. As an application, an alternative solution to the conjugacy
problem in is given
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