Bounded Generation for $SL_n(\Lambda)$
Let $\Lambda$ be an order in a division algebra over a number field. We prove, under some conditions, that $SL_3(\Lambda)$ is boundedly generated by elementary matrices.
math.GR↗
arXiv subjects
Chen Meiri
Publications and source records attributed to Chen Meiri.
Conjugacy width in uniform higher rank arithmetic groups of orthogonal type
We study widths of conjugacy classes in anisotropic higher rank $S$-arithmetic groups of orthogonal type. Assuming the GRH, we prove that many such groups have bounded conjugacy width. For example, this holds if the degree is greater or equal to 17 and $S$ contains a non-archimedean place. To the best of our knowledge, this is the first boundedness result proved for anisotropic groups. The proof uses ideas from the Congruence Subgroup Problem. In particular, we define and compute a non standard version of the metaplectic kernel. Conversely, we prove that a quantitative bound on the width of conjugacy classes implies the CSP. The machinery we develop can also be used for other width questions. For example, in \cite{AM25} we prove, unconditional on GRH, new cases of bounded generation of arithmetic groups.
math.GR↗
Commutators in $\mathrm{SL}_2$ and Markoff surfaces I
We show that the commutator equation over $\mathrm{SL}_2(\mathbb{Z})$ satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for $ \mathrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$. The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.
math.NT↗
Higher rank arithmetic groups which are not invariably generated
It was conjectured in [KLS14] that for arithmetic groups, Invariable Generation is equivalent to the Congruence Subgroup Property. In view of the famous Serre conjecture this would imply that higher rank arithmetic groups are invariably generated. In this paper we prove that some higher rank arithmetic groups are not invariably generated.
math.GR↗
Is being a higher rank lattice a first order property?
We show that there is a sentence $φ$ in the first order language of groups such that a finitely generated group $Γ$ satisfies $φ$ if and only if $Γ$ is isomorphic to a group of the form $\mathrm{PSL}_n(O)$, where $n \geq 3$ and $O$ is a ring of $S$-integers in a number field.
math.GR↗
On the model theory of higher rank arithmetic groups
Let $Γ$ be a centerless irreducible higher rank arithmetic lattice in characteristic zero. We prove that if $Γ$ is either non-uniform or is uniform of orthogonal type and dimension at least 9, then $Γ$ is bi-interpretable with the ring $\mathbb{Z}$ of integers. It follows that the first order theory of $Γ$ is undecidable, that all finitely generated subgroups of $Γ$ are definable, and that $Γ$ is characterized by a single first order sentence among all finitely generated groups.
math.GR↗
Some Orbits of Free Words that are Determined by Measures on Finite Groups
Every word in a free group $F$ induces a probability measure on every finite group in a natural manner. It is an open problem whether two words that induce the same measure on every finite group, necessarily belong to the same orbit of $\mathrm{Aut}F$. A special case of this problem, when one of the words is the primitive word $x$, was settled positively by the third author and Parzanchevski [arXiv:1202.3269]. Here we extend this result to the case where one of the words is $x^d$ or $\left[x,y\right]^{d}$ for an arbitrary $d\in\mathbb{Z}$.
math.GR↗
Words have bounded width in $SL(n,\mathbb{Z})$
We prove two results about width of words in $SL_n(\mathbb{Z})$. The first is that, for every $n \geq 3$, there is a constant $C(n)$ such that the width of any word in $SL_n(\mathbb{Z})$ is less than $C(n)$. The second result is that, for any word $w$, if $n$ is big enough, the width of $w$ in $SL_n(\mathbb{Z})$ is at most 87.
math.GR↗
The Markoff Group of Transformations in Prime and Composite Moduli
The Markoff group of transformations is a group $Γ$ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation $x^{2}+y^{2}+z^{2}=xyz$. The fundamental strong approximation conjecture for the Markoff equation states that for every prime $p$, the group $Γ$ acts transitively on the set $X^{*}\left(p\right)$ of non-zero solutions to the same equation over $\mathbb{Z}/p\mathbb{Z}$. Recently, Bourgain, Gamburd and Sarnak proved this conjecture for all primes outside a small exceptional set. In the current paper, we study a group of permutations obtained by the action of $Γ$ on $X^{*}\left(p\right)$, and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that $Γ$ acts transitively also on the set of non-zero solutions in a big class of composite moduli. Our result is also related to a well-known theorem of Gilman, stating that for any finite non-abelian simple group $G$ and $r\ge3$, the group $\mathrm{Aut}\left(F_{r}\right)$ acts on at least one $T_{r}$-system of $G$ as the alternating or symmetric group. In this language, our main result translates to that for most primes $p$, the group $\mathrm{Aut}\left(F_{2}\right)$ acts on a particular $T_{2}$-system of $\mathrm{PSL}\left(2,p\right)$ as the alternating or symmetric group.
math.NT↗
First order rigidity of non-uniform higher rank arithmetic lattices
If $Γ$ is an irreducible non-uniform higher-rank characteristic zero arithmetic lattice (for example, $SL_n(\mathbb{Z})$, $n \geq 3$) and $Λ$ is a finitely generated group that is elementarily equivalent to $Γ$, then $Λ$ is isomorphic to $Γ$.
math.GR↗
Maximal subgroups of SL(n, Z)
We establish the existence of maximal subgroups of various diferent natures in SL(n,Z). In particular, we prove that there are continuously many maximal subgroups, we provide a maximal subgroup whose action on the projective space has no dense orbits, and we produce a faithful primitive permutation representation of PSL(n,Z) which is not 2-transitive.
math.GR↗
Generating pairs for finite index subgroups of sl(n, z)
Let $n \ge 3$. We positively answer a question of Lubotzky and prove that every finite index subgroup of SL(n, Z) contains a finite index subgroup which is generated by two elements.
math.GR↗
On the non-amenability of the reflective quotient \Rmnum{1}: The rational case
Let O(f,Z) be the integral orthogonal group of an integral quadratic form f of signature (n,1). Let R(f,Z) be the subgroup of O(f,Z) generated by all hyperbolic reflections. Vinberg proved that if n > 29 then the reflective quotient O(f,Z)/R(f,Z) is infinite. In this note we generalize Vinberg's theorem and prove that if n > 91 then O(f,Z)/R(f,Z) contains a non-abelian free group (and thus it is not amenable).
math.GR↗
Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions
We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n-1,1) is "thin", namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg's theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for n_F_(n-1) are thin.
math.GR↗
Sieve methods in group theory II: The Mapping Class Group
We prove that the set of non-pseudo-Anosov elements in the Torelli group is exponentially small.
math.GR↗
Sieve methods in group theory I: Powers in Linear groups
A general sieve method for groups is formulated. It enables one to "measure" subsets of a finitely generated group. As an application we show that if $Γ$ is a finitely generated non virtually-solvable linear group of characteristic zero then the set of proper powers in $Γ$ is exponentially small. This is a far reaching strengthening of the main result of \cite{HKLS}.
math.GR↗
Sieve methods in group theory \Rmnum{3}: $\aut(F_n)$
Let $π:\aut(F_n)\rightarrow \aut(\Z^n)$ be the epimorphism induced by the isomorphism $\Z^n \cong F_n/F_n'$ and define $\mathcal{T}_n:=\kerπ$. We prove that the subset of $\mathcal{T}_n$ consists of all non-iwip and all non-hyperbolic elements is exponentially small.
math.GR↗
On conjugacy growth of linear groups
We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GL_d have uniform exponential conjugacy growth and in fact that the number of distinct polynomials arising as characteristic polynomials of the elements of the ball of radius n for the word metric has exponential growth rate bounded away from 0 in terms of the dimension d only.
math.GR↗