Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNielsen equivalence in Gupta-Sidki groups
78
0
0.0
(
0
)
Authors
Aglaia Myropolska
Ask ChatGPT about the research
No Arabic abstract
For a group $G$ generated by $k$ elements, the Nielsen equivalence classes are defined as orbits of the action of $operatorname{Aut} F_k$, the automorphism group of the free group of rank $k$, on the set of generating $k$-tuples of $G$. Let $pgeq 3$ be prime and $G_p$ the Gupta-Sidki $p$-group. We prove that there are infinitely many Nielsen equivalence classes on generating pairs of $G_p$.
rate research
Read More
We exhibit explicit infinite families of finitely presented, Kazhdan, simple groups that are pairwise not measure equivalent. These groups are lattices acting on products of buildings. We obtain the result by studying vanishing and non-vanishing of their $L^2$-Betti numbers.
We describe a family of finitely presented groups which are quasi-isometric but not bilipschitz equivalent. The first such examples were described by the first author and are the lamplighter groups $F wr mathbb{Z}$ where $F$ is a finite group; these groups are finitely generated but not finitely presented. The examples presented in this paper are higher rank generalizations of these lamplighter groups and include groups that are of type $F_n$ for any $n$.
The object of this expository work is to try to unveil the topological/geometric intuition behind the theory of free groups and their automorphism and outer automorphism groups. The method we follow is to focus on a series of problems in the study of free groups, and use the solutions of those problems to motivate topological/geometric tools. We do not aim to write down proofs which minimize the number of alphanumeric characters. We instead strive to write down proofs which maximize the development of broadly applicable geometric tools. In Part I we study problems solved by Nielsen and Whitehead in the 1920s and 1930s, but we approach these problems from a modern topological/geometric viewpoint, and we formulate their solutions so as to motivate modern tools, including marked graphs, the outer space of a free group, and fold paths in outer space.
Divergence functions of a metric space estimate the length of a path connecting two points $A$, $B$ at distance $le n$ avoiding a large enough ball around a third point $C$. We characterize groups with non-linear divergence functions as groups having cut-points in their asymptotic cones. By Olshanskii-Osin-Sapir, that property is weaker than the property of having Morse (rank 1) quasi-geodesics. Using our characterization of Morse quasi-geodesics, we give a new proof of the theorem of Farb-Kaimanovich-Masur that states that mapping class groups cannot contain copies of irreducible lattices in semi-simple Lie groups of higher ranks. It also gives a generalization of the result of Birman-Lubotzky-McCarthy about solvable subgroups of mapping class groups not covered by the Tits alternative of Ivanov and McCarthy. We show that any group acting acylindrically on a simplicial tree or a locally compact hyperbolic graph always has many periodic Morse quasi-geodesics (i.e. Morse elements), so its divergence functions are never linear. We also show that the same result holds in many cases when the hyperbolic graph satisfies Bowditchs properties that are weaker than local compactness. This gives a new proof of Behrstocks result that every pseudo-Anosov element in a mapping class group is Morse. On the other hand, we conjecture that lattices in semi-simple Lie groups of higher rank always have linear divergence. We prove it in the case when the $mathbb{Q}$-rank is 1 and when the lattice is $SL_n(mathcal{O}_S)$ where $nge 3$, $S$ is a finite set of valuations of a number field $K$ including all infinite valuations, and $mathcal{O}_S$ is the corresponding ring of $S$-integers.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
