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Nattalie Tamam

Publications and source records attributed to Nattalie Tamam.

10 recordsLinked to original sources
arXiv2025

Geometric interpretation of quantitative instability

Given a real algebraic group $G$ acting on a linear space $V$, a vector $v\in V$ is called unstable if $0\in \overline{Gv}-Gv$, where the closure is taken with respect to the Zariski topology. A fundamental theorem of Kempf in geometric invariant theory states that $v$ is unstable if and only if there is a one-parameter subgroup $A$ of $G$ such that $Av$ is unstable. Assuming $G$ is a semisimple real algebraic $\mathbb{Q}$-group, we give a new proof to this result using a geometric interpretation of the setting. In the process, we also give a new proof of an effective version of this result by Shah and Yang. Our interpretation involves relating the length of vectors under a linear action to convex functions on certain $\cat$-spaces, and bound the later from below by Busemann functions.

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arXiv2024

On weighted singular vectors for multiple weights

We introduce the notion of weighted singular vectors and weighted uniform exponent with respect to a set of weights. We prove invariance of these exponents for affine subspaces and submanifolds inside those affine subspaces. For certain analytic submanifolds, we show that there are totally irrational vectors with high weighted uniform exponent, extending the previously known existence results. Moreover, we show existence and non-existence of non-obvious divergence orbits for certain cones.

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arXiv2023

On Topologically Big Divergent Trajectories

We study the behavior of $A$-orbits in $G/Γ$, when $G$ is a semisimple real algebraic $\mathbb{Q}$-group, $Γ$ is a non-uniform arithmetic lattice, and $A$ is a torus of dimension $\geq\operatorname{rank}_\mathbb{Q}(Γ)$. We show that every divergent trajectory of $A$ diverges due to a purely algebraic reason, % has a simple algebraic description. solving a longlasting conjecture of Weiss. In addition, we examine the intersections of $A$-orbits and show that in many cases every $A$-orbit intersects every deformation retract $X\subseteq G/Γ$. This solves the questions raised by Pettet and Souto. The proofs use algebraic and differential topology, as well as algebraic group theory.

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arXiv2022

Property tau in positive characteristic

We prove a quantitative equidistribution statement for certain adelic homogeneous subsets in positive characteristic. As an application, we describe a proof of property tau for arithmetic groups in this context.

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arXiv2021

Distribution of orbits of geometrically finite groups acting on null vectors

We study the distribution of non-discrete orbits of geometrically finite groups in $\operatorname{SO}(n,1)$ acting on $\mathbb{R}^{n+1}$, and more generally on the quotient of $\operatorname{SO}(n,1)$ by a horospherical subgroup. Using equidistribution of horospherical flows, we obtain both asymptotics for the distribution of orbits for the action of general geometrically finite groups, and we obtain quantitative statements with additional assumptions.

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arXiv2021

Effective equidistribution of horospherical flows in infinite volume rank one homogeneous spaces

We prove effective equidistribution of horospherical flows in $\operatorname{SO}(n,1)^\circ / Γ$ when $Γ$ is geometrically finite and the frame flow is exponentially mixing for the Bowen-Margulis-Sullivan measure. We also discuss settings in which such an exponential mixing result is known to hold. As part of the proof, we show that the Patterson-Sullivan measure satisfies some friendly-like properties when $Γ$ is geometrically finite.

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arXiv2021

Existence of Non-Obvious Divergent Trajectories in homogeneous spaces

We prove a modified version for a conjecture of Weiss from 2004. Let $G$ be a semisimple real algebraic group defined over $\mathbb{Q}$, $Γ$ be an arithmetic subgroup of $G$. A trajectory in $G/Γ$ is divergent if eventually it leaves every compact subset, and is obvious divergent if there is a finite collection of algebraic data which cause the divergence. Let $A$ be a diagonalizable subgroup of $G$ of positive dimension. We show that if the projection of $A$ to any $\mathbb{Q}$-factor of $G$ is of small enough dimension (relatively to the $\mathbb{Q}$-rank of the $\mathbb{Q}$-factor), then there are non-obvious divergent trajectories for the action of $A$ on $G/Γ$.

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arXiv2019

Divergent Trajectories in Arithmetic Homogeneous Spaces of Rational Rank Two

Let $G$ be a real algebraic group defined over $\mathbb{Q}$, $Γ$ be an arithmetic subgroup of $G$, and $T$ be a maximal $\mathbb{R}$-split torus. A trajectory in $G/Γ$ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which account for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in $T$, we study the existence of non-obvious divergent trajectories under its action in $G/Γ$. We get a sufficient condition for the existence of a non-obvious divergence trajectory in the general case, and a full classification under the assumption that $\mbox{rank}_{\mathbb{Q}}G=\mbox{rank}_{\mathbb{R}}G=2$.

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arXiv2018

Hausdorff dimension of weighted singular vectors

Let $w=(w_1, w_2)$ be a pair of positive real numbers with $w_1+w_2=1$ and $w_1\ge w_2$. We show that the set of $w$-weighted singular vectors in $\mathbb R^2$ has Hausdorff dimension $2- \frac{1}{1+w_1}$. This extends the previous work of Yitwah Cheung on the Hausdorff dimension of the usual (unweighted) singular vectors in $\mathbb R^2$.

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