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Michael Larsen

Publications and source records attributed to Michael Larsen.

At least 55 records · Page 3Linked to original sources
arXiv2017

Categorical measures for finite group actions

Given a variety with a finite group action, we compare its equivariant categorical measure, that is, the categorical measure of the corresponding quotient stack, and the categorical measure of the extended quotient. Using weak factorization for orbifolds, we show that for a wide range of cases, these two measures coincide. This implies, in particular, a conjecture of Galkin and Shinder on categorical and motivic zeta-functions of varieties. We provide examples showing that, in general, these two measures are not equal. We also give an example related to a conjecture of Polishchuk and Van den Bergh, showing that a certain condition in this conjecture is indeed necessary.

math.AG↗
arXiv2017

Character Levels and Character Bounds

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig's label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large.

math.RT↗
arXiv2017

Waring's problem for unipotent algebraic groups

In this paper, we formulate an analogue of Waring's problem for an algebraic group $G$. At the field level we consider a morphism of varieties $f\colon \mathbb{A}^1\to G$ and ask whether every element of $G(K)$ is the product of a bounded number of elements $f(\mathbb{A}^1(K)) = f(K)$. We give an affirmative answer when $G$ is unipotent and $K$ is a characteristic zero field which is not formally real. The idea is the same at the integral level, except one must work with schemes, and the question is whether every element in a finite index subgroup of $G(\mathcal{O})$ can be written as a product of a bounded number of elements of $f(\mathcal{O})$. We prove this is the case when $G$ is unipotent and $\mathcal{O}$ is the ring of integers of a totally imaginary number field.

math.NT↗
arXiv2017

Diophantine stability

If $V$ is an irreducible algebraic variety over a number field $K$, and $L$ is a field containing $K$, we say that $V$ is diophantine-stable for $L/K$ if $V(L) = V(K)$. We prove that if $V$ is either a simple abelian variety, or a curve of genus at least one, then under mild hypotheses there is a set $S$ of rational primes with positive density such that for every $\ell \in S$ and every $n \ge 1$, there are infinitely many cyclic extensions $L/K$ of degree $\ell^n$ for which $V$ is diophantine-stable. We use this result to study the collection of finite extensions of $K$ generated by points in $V(\bar{K})$.

math.NT↗
arXiv2017

Words, Hausdorff dimension and randomly free groups

We study fibers of word maps in finite, profinite, and residually finite groups. Our main result is that, for any word w in the free group on d generators, there exists $ε> 0$ such that if G is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map $w\colon G^d \to G$ have Hausdorff dimension at most $d-ε$. We conclude that the profinite completion of a group G as above satisfies no probabilistic identity. It is therefore randomly free; namely, for any d > 0, the probability that d randomly chosen elements freely generate a free subgroup of G is 1. This solves an open problem of Dixon, Pyber, Seress, and Shalev. Additional applications and related results are also established. For example, combining our results with recent results of Bors, we conclude that a profinite group in which the set of elements of finite odd order has positive measure has an open prosolvable subgroup. This may be regarded as a probabilistic version of the Feit-Thompson theorem.

math.GR↗
arXiv2016

Character varieties and actions on products of trees

It is well known that surface groups admit free and proper actions on finite products of infinite valence trees. In this note, we address the question of whether there can be a free and proper action on a finite product of bounded valence trees. We provide some obstructions and an arithmetic criterion for existence. The bulk of the paper is devoted to an approach to verifying the arithmetic criterion by studying the character variety of certain surface groups over fields of positive characteristic. The methods may be useful for attempting to determine when groups admit good linear representations in other contexts.

math.GR↗
arXiv2015

Adelic openness without the Mumford-Tate conjecture

Let $X$ be a non-singular projective variety over a number field $K$, $i$ a non-negative integer, and $V_{\A}$, the etale cohomology of $\bar X$ with coefficients in the ring of finite adeles $\A_f$ over $\Q$. Assuming the Mumford-Tate conjecture, we formulate a conjecture (Conjecture 1.2) describing the largeness of the image of the absolute Galois group $G_K$ in $H(\A_f)$ under the adelic Galois representation $ρ_{\A}: G_K -> \Aut(V_{\A})=\GL_n(\A_f)$, where $H$ is the Hodge group. The motivating example is a celebrated theorem of Serre, which asserts that if $X$ is an elliptic curve without complex multiplication over $\bar K$ and $i=1$, then $ρ_{\A}(G_K)$ is an open subgroup of $\GL_2(\hat \Z)\subset \GL_2(\A_f)$. We state and in some cases prove a weaker conjecture which does not require Mumford-Tate but which, together with Mumford-Tate, implies Conjecture 1.2. We also relate our conjectures to Serre's conjectures on maximal motives.

math.NT↗
arXiv2015

A probabilistic Tits alternative and probabilistic identities

We introduce the notion of a probabilistic identity of a residually finite group. We prove that a finitely generated linear group satisfies a probabilistic identity if and only if it is virtually solvable. As an application, we prove a probabilistic variant of the Tits alternative: let G be the profinite completion of a finitely generated linear group. Then either G is virtually solvable, or for any positive integer n, with probability one, n independent, uniformly distributed elements of G freely generate a free subgroup of G of rank n.

math.GR↗
arXiv2015

Functoriality and the Inverse Galois Problem

We prove that there are infinitely many finite simple groups of symplectic Lie type, of any specified characteristic and rank, which appear as Galois groups over the field of rational numbers. This generalizes a result of Wiese, which inspired this paper.

math.NT↗
arXiv2015

Arithmetic quotients of the mapping class group

To every $Q$-irreducible representation $r$ of a finite group $H$, there corresponds a simple factor $A$ of $Q[H]$ with an involution $τ$. To this pair $(A,τ)$, we associate an arithmetic group $Ω$ consisting of all $(2g-2)\times (2g-2)$ matrices over a natural order of $A^{op}$ which preserve a natural skew-Hermitian sesquilinear form on $A^{2g-2}$. We show that if $H$ is generated by less than $g$ elements, then $Ω$ is a virtual quotient of the mapping class group $Mod(Σ_g)$, i.e. a finite index subgroup of $Ω$ is a quotient of a finite index subgroup of $\Mod(Σ_g)$. This shows that the mapping class group has a rich family of arithmetic quotients (and "Torelli subgroups") for which the classical quotient $Sp(2g, Z)$ is just a first case in a list, the case corresponding to the trivial group $H$ and the trivial representation. Other pairs of $H$ and $r$ give rise to many new arithmetic quotients of $Mod(Σ_g)$ which are defined over various (subfields of) cyclotomic fields and are of type $Sp(2m), SO(2m,2m),$ and $SU(m,m)$ for arbitrarily large $m$.

math.GT↗
arXiv2014

Type A Images of Galois Representations and Maximality

Given a compatible system of n-dimension l-adic Galois representations arising from étale cohomology of any complete, non-singular variety, we prove that for sufficiently large prime l, type A Galois image (in the Cartan-Killing classification) is in some sense maximal in its Zariski closure in GL(n).

math.NT↗
arXiv2014

Representation varieties of Fuchsian groups

We estimate the dimension of varieties of the form Hom(F,G) where F is a Fuchsian group and G is a simple real algebraic group, answering along the way a question of I. Dolgachev.

math.AG↗
arXiv2014

The Waring problem for Lie groups and Chevalley groups

The classical Waring problem deals with expressing every natural number as a sum of g(k) k-th powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given non-trivial word w. In this paper we study this problem for Lie groups and Chevalley groups over infinite fields. We show that for a fixed non-trivial word w and for a classical connected real compact Lie group G of sufficiently large rank we have w(G)^2=G, namely every element of G is a product of 2 values of w. We prove a similar result for non-compact Lie groups of arbitrary rank, arising from Chevalley groups over R or over a p-adic field. We also study this problem for Chevalley groups over arbitrary infinite fields, and show in particular that every element in such a group is a product of two squares.

math.GR↗
arXiv2014

A Finitary Hasse Principle for Diagonal Curves

We prove a Hasse principle for solving equations of the form ax+by+cz=0 where x, y, z belong to a given finite index subgroup of the multiplicative group of rational numbers. From this we deduce a Hasse principle for diagonal curves over fields of algebraic numbers with finitely generated Galois group.

math.NT↗
arXiv2013

Linear groups with Borel's property

When does Borel's theorem on free subgroups of semisimple groups generalize to other groups? We initiate a systematic study of this question and find positive and negative answers for it. In particular, we fully classify fundamental groups of surfaces and von Dyck groups that satisfy Borel's theorem. Further, as a byproduct of this theory, we make headway on a question of Breuillard, Green, Guralnick, and Tao concerning double word maps.

math.GR↗
arXiv2013

A Refined Waring Problem for Finite Simple Groups

Let v and w be nontrivial words in two free groups. We prove that, for all sufficiently large finite non-abelian simple groups G, there exist subsets C of v(G) and D of w(G) of size such that every element of G can be realized in at least one way as the product of an element of C and an element of D and the average number of such representations is O(log |G|). In particular, if w is a fixed nontrivial word and G is a sufficiently large finite non-abelian simple group, then w(G) contains a thin base of order 2. This is a non-abelian analogue of a result of Van Vu for the classical Waring problem. Further results concerning thin bases of G of order 2 are established for any finite group and for any compact Lie group G.

math.GR↗
arXiv2013

Rational curves on quotients of abelian varieties by finite groups

In [3], it is proved that the quotient of an abelian variety $A$ by a finite order automorphism $g$ is uniruled if and only if some power of $g$ satisfies a numerical condition $0<\age(g^k)<1$. In this paper, we show that $\age(g^k)=1$ is enough to guarantee that $A/\langle g\rangle$ has at least one rational curve.

math.AG↗