Research

I am interested generally in actions of groups on manifolds, and the rigidity phenomena that may arise. A popular instance of this is the Zimmer Programme, which aims to classify that possible actions that lattices in higher rank semisimple groups can have on manifolds, using tools and techniques from algebra, geometry, and dynamics.

Occasionally I am also interested in studying group algebras and their analytic and cohomological properties.

Publications

1. (In preparation) - Zimmer's conjecture for lattices in semisimple and S-arithmetic groups.

2. Equidistribution of Hecke neighbours and continued fraction patterns - with Menny Akka, 2026.

   Abstract

   We prove uniform tightness of Hecke neighbours of any probability measure of positive entropy that is invariant and ergodic under the diagonal flow. As an application, we confirm a conjecture of Einsiedler concerning a homogeneous dynamics analogue of an equidistribution theorem of Johnson and Rudolph on the torus. We also obtain applications to continued-fraction patterns in arithmetic sequences and to the \(p\)-adic Littlewood conjecture. Our proof studies escape of mass using dynamics on the boundary of the Chabauty compactification of the space of lattices, together with averaging operators defined by leafwise measures along horospherical subgroups.

3. Discontinuous actions of rational special linear groups on manifolds (arXiv), 2026.

   Abstract

   For every $n \geq 1$, we construct continuum-many pairwise non-conjugate actions of $\operatorname{SL}(n+1,\mathbb{Q})$ by homeomorphisms on the n-sphere. None of these actions extend to a continuous action of $\operatorname{SL}(n+1,\mathbb{R})$. Our construction refines a blow-up procedure of Bowden and Mann, allowing the stabiliser action inside each blown-up disc to be prescribed. We motivate studying these actions and record some related observations.

4. Actions of lattices in p-adic and S-arithmetic groups on manifolds (arXiv), 2026.

   Abstract

   We prove that an action by $C^1$-diffeomorphisms of a lattice in a simple $p$-adic group on a compact manifold is finite, provided that the dimension of the manifold is less than the rank of the group. We extend this statement to lattices in totally disconnected $S$-arithmetic groups, where the critical dimension is the maximal rank of its simple factors. This uses the machinery developed by Brown, Fisher, and Hurtado.

5. Impact Hamiltonian systems and polygonal billiards (arXiv) - with L. Becker, S. Elliott, B. Firester, M. Pnueli, V. Rom-Kedar, 2018.

   In Hamiltonian Systems: Dynamics, Analysis, Applications, edited by Albert Fathi, Philip J. Morrison, Tere M-Seara, Sergei Tabachnikov, MSRI publications, Vol 72, 2024.

   This work was done in part during the 2018 ISSI in the Weizmann Institute of Science.

   Abstract

   The dynamics of a beam held on a horizontal frame by springs and bouncing off a step is described by a separable two degrees of freedom Hamiltonian system with impacts that respect, point wise, the separability symmetry. The energy in each degree of freedom is preserved, and the motion along each level set is conjugated, via action angle coordinates, to a geodesic flow on a flat two-dimensional surface in the four dimensional phase space. Yet, for a range of energies, these surfaces are not the simple Liouville-Arnold tori - these are tori of genus two, thus the motion on them is not conjugated to simple rotations. Namely, even though energy is not transferred between the two degrees of freedom, the impact system is quasi-integrable and is not of the Liouville-Arnold type. In fact, for each level set in this range, the motion is conjugated to the well studied and highly non-trivial dynamics of directional motion in L-shaped billiards, where the billiard area and shape as well as the direction of motion vary continuously on iso-energetic level sets. Return maps to Poincaré section of the flow are shown to be conjugated, on each level set, to interval exchange maps which are computed, up to quadratures, in the general nonlinear case and explicitly for the case of two linear oscillators bouncing off a step. It is established that for any such oscillator-step system there exist step locations for which some of the level sets exhibit motion which is neither periodic nor ergodic. Changing the impact surface by introducing additional steps, staircases, strips and blocks from which the particle is reflected, leads to iso-energy surfaces that are foliated by families of genus-k level set surfaces, where the number and order of families of genus k depend on the energy.