Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOne Variable Equations in Torsion-Free Hyperbolic Groups
187
0
0.0
(
0
)
Authors
Abderezak Ould Houcine
Ask ChatGPT about the research
No Arabic abstract
Let $Gamma$ be a torsion-free hyperbolic group. We show that the set of solutions of any system of equations with one variable in $Gamma$ is a finite union of points and cosets of centralizers if and only if any two-generator subgroup of $Gamma$ is free.
rate research
Read More
We show that any nonabelian free group $F$ of finite rank is homogeneous; that is for any tuples $bar a$, $bar b in F^n$, having the same complete $n$-type, there exists an automorphism of $F$ which sends $bar a$ to $bar b$. We further study existential types and we show that for any tuples $bar a, bar b in F^n$, if $bar a$ and $bar b$ have the same existential $n$-type, then either $bar a$ has the same existential type as a power of a primitive element, or there exists an existentially closed subgroup $E(bar a)$ (resp. $E(bar b)$) of $F$ containing $bar a$ (resp. $bar b$) and an isomorphism $sigma : E(bar a) to E(bar b)$ with $sigma(bar a)=bar b$. We will deal with non-free two-generated torsion-free hyperbolic groups and we show that they are $exists$-homogeneous and prime. This gives, in particular, concrete examples of finitely generated groups which are prime and not QFA.
We show that any one-relator group $G=F/langlelangle wranglerangle$ with torsion is coherent -- i.e., that every finitely generated subgroup of $G$ is finitely presented -- answering a 1974 question of Baumslag in this case.
A famous result of Hall asserts that the multiplication and exponentiation in finitely generated torsion free nilpotent groups can be described by rational polynomials. We describe an algorithm to determine such polynomials for all torsion free nilpotent groups of given Hirsch length. We apply this to determine the Hall polynomials for all such groups of Hirsch length at most 7.
We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, as shortlex geodesic words (or any regular set of quasigeodesic normal forms), is an EDT0L language whose specification can be computed in NSPACE$(n^2log n)$ for the torsion-free case and NSPACE$(n^4log n)$ in the torsion case. Furthermore, in the presence of quasi-isometrically embeddable rational constraints, we show that the full set of solutions to systems of equations in a hyperbolic group remains EDT0L. Our work combines the geometric results of Rips, Sela, Dahmani and Guirardel on the decidability of the existential theory of hyperbolic groups with the work of computer scientists including Plandowski, Je.z, Diekert and others on PSPACE algorithms to solve equations in free monoids and groups using compression, and involves an intricate language-theoretic analysis.
In this paper we determine the torsion free rank of the group of endotrivial modules for any finite group of Lie type, in both defining and non-defining characteristic. On our way to proving this, we classify the maximal rank $2$ elementary abelian $ell$-subgroups in any finite group of Lie type, for any prime $ell$, which may be of independent interest.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
