We show that, in a weakly regular p-adic Lie group G, the subgroup ${\mathit{G}}_{\mathit{u}}$ spanned by the one-parameter subgroups of G admits a Levi decomposition. As a consequence, there exists a regular open subgroup of G which contains ${\mathit{G}}_{\mathit{u}}$

Let Γ be a nonelementary Kleinian group and $\mathit{H}<\mathrm{\Gamma }$ be a finitely generated, proper subgroup. We prove that if Γ has finite covolume, then the profinite completions of H and Γ are not isomorphic. If H has finite index in Γ, then there is a finite group onto which H maps but Γ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circle of ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, for example, limit groups. We construct new examples of profinitely rigid groups, including the fundamental group of the hyperbolic 3-manifold $\mathrm{Vol}(3)$ and that of the 4-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in $\mathrm{PSL}(2,\mathbb{C})$ is profinitely rigid, then so is its normalizer in $\mathrm{PSL}(2,\mathbb{C})$.

Given two algebraic groups G, H over a field k, we investigate the representability of the functor of morphisms (of schemes) $\mathbf{Hom}(\mathit{G},\mathit{H})$ and the subfunctor of homomorphisms (of algebraic groups) ${\mathbf{Hom}}_{\mathrm{gp}}(\mathit{G},\mathit{H})$. We show that $\mathbf{Hom}(\mathit{G},\mathit{H})$ is represented by a group scheme, locally of finite type, if the k-vector space $\mathcal{O}(\mathit{G})$ is finite-dimensional; the converse holds if H is not étale. When G is linearly reductive and H is smooth, we show that ${\mathbf{Hom}}_{\mathrm{gp}}(\mathit{G},\mathit{H})$ is represented by a smooth scheme M; moreover, every orbit of H acting by conjugation on M is open.

We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; that is, over one-variable function fields over complete discretely valued fields. We provide conditions on the group and the semi-global field under which the local-global principle holds, and we compute the obstruction to the local-global principle in certain classes of examples. Using our description of the obstruction, we give the first example of a semisimple simply connected group over a semi-global field where the local-global principle fails. Our methods include patching and R-equivalence.

We study the action of the derived Hecke algebra in the setting of dihedral weight one forms and prove a conjecture of the second- and fourth- named authors relating this action to certain Stark units associated to the symmetric square L-function. The proof exploits the theta correspondence between various Hecke modules as well as the ideas of Merel and Lecouturier on higher Eisenstein elements.

After recalling some basic facts about F-wound commutative unipotent algebraic groups over an imperfect field F, we study their regular integral models over Dedekind schemes of positive characteristic and compute the group of isomorphism classes of torsors of one-dimensional groups.

In the late seventies, Sullivan showed that, for a convex cocompact subgroup Γ of ${\mathrm{SO}}^{\circ }(\mathit{n},1)$ with critical exponent $\mathit{\delta }>0$, any Γ-conformal measure on $\partial {\mathbb{H}}^{\mathit{n}}$ of dimension δ is necessarily supported on the limit set Λ and that the conformal measure of dimension δ exists uniquely. We prove an analogue of this theorem for any Zariski dense Anosov subgroup Γ of a connected semisimple real algebraic group G of rank at most 3. We also obtain the local mixing for generalized BMS measures on $\mathrm{\Gamma }\setminus \mathit{G}$ including Haar measures.

Let F be a p-adic field, that is, a finite extension of ${\mathbb{Q}}_{\mathit{p}}$. Let D be a finite dimensional division algebra over F, and let ${\mathrm{SL}}_1(\mathit{D})$ be the group of elements of reduced norm 1 in D. Prasad and Raghunathan proved that ${\mathit{H}}^2({\mathrm{SL}}_1(\mathit{D}),\mathbb{R}/\mathbb{Z})$ is a cyclic p-group whose order is bounded from below by the number of p-power roots of unity in F, unless D is a quaternion algebra over ${\mathbb{Q}}_2$. In this paper we give an explicit upper bound for the order of ${\mathit{H}}^2({\mathrm{SL}}_1(\mathit{D}),\mathbb{R}/\mathbb{Z})$ for $\mathit{p}\ge 5$ and determine ${\mathit{H}}^2({\mathrm{SL}}_1(\mathit{D}),\mathbb{R}/\mathbb{Z})$ precisely when F is cyclotomic, $\mathit{p}\ge 19$, and the degree of D is not a power of p.

Let F be a nonarchimedean local field of residual characteristic $\mathit{p}\ne 2$. Let G be a (connected) reductive group that splits over a tamely ramified field extension of F. We show that a construction analogous to Yu’s construction of complex supercuspidal representations yields smooth, irreducible, cuspidal representations over an arbitrary algebraically closed field R of characteristic different from p. Moreover, we prove that this construction provides all smooth, irreducible, cuspidal R-representations if p does not divide the order of the Weyl group of G.

We prove a myriad of results related to the stabilizer in an algebraic group G of a generic vector in a representation V of G over an algebraically closed field k. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of G and the group $\mathit{G}(\mathit{k})$ of k-points. For G simple and V faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those G and V for which the stabilizer in general position is smooth, or $dim\mathit{V}/\mathit{G}<dim\mathit{G}$, or there is a $\mathit{v}\in \mathit{V}$ whose stabilizer in G is trivial.

We prove an effective variant of the Kazhdan–Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a nontrivial intersection with a small r-neighborhood of the identity is at most $\mathit{\beta }{\mathit{r}}^{\mathit{\delta }}$ for some explicit constants $\mathit{\beta },\mathit{\delta }>0$ depending only on the group. This is a consequence of a key convolution inequality. We deduce that vanishing at infinity of injectivity radius implies finiteness of volume. Further applications are the compactness of the space of discrete stationary random subgroups and a novel proof of the fact that all lattices in semisimple groups are weakly cocompact.

We show that a reductive group scheme over a base scheme S admits a faithful linear representation if and only if its radical torus is isotrivial; that is, it splits after a finite étale cover.

Abert, Gelander, and Nikolov [AGR17] conjectured that the number of generators $\mathit{d}(\mathrm{\Gamma })$ of a lattice Γ in a high rank simple Lie group H grows sublinearly with $\mathit{v}=\mathit{\mu }(\mathit{H}/\mathrm{\Gamma })$, the co-volume of Γ in H. We prove this for nonuniform lattices in a very strong form, showing that for 2-generic such Hs, $\mathit{d}(\mathrm{\Gamma })={\mathit{O}}_{\mathit{H}}(log\mathit{v}/loglog\mathit{v})$, which is essentially optimal. Although we cannot prove a new upper bound for uniform lattices, we will show that for such lattices one cannot expect to achieve a better bound than $\mathit{d}(\mathrm{\Gamma })=\mathit{O}(log\mathit{v})$.

In this work, we introduce and study the notion of local randomness for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional dimension condition on the volume of small balls, and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we develop a Littlewood–Paley decomposition and explore its connection to the existence of a spectral gap for random walks. Moreover, under the dimension condition alone, we prove a multi-scale entropy gain result à la Bourgain–Gamburd and Tao.

We give a classification of special reductive groups over arbitrary fields that improves a theorem of M. Huruguen.

Given a reductive group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat–Tits building into the analytic space associated with the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.

This paper studies residual finiteness of lattices in the universal cover of $\mathrm{PU}(2,1)$ and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in $\mathrm{PU}(2,1)$ or a finite covering of it. First, we prove that certain lattices in the universal cover of $\mathrm{PU}(2,1)$ are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in $\mathrm{PU}(2,1)$ to construct smooth projective surfaces that are not birationally equivalent to a smooth compact ball quotient but whose fundamental group is a torsion-free cocompact lattice in $\mathrm{PU}(2,1)$.

We give a simple proof of the centrality of the congruence subgroup kernel in the higher rank isotropic case.

Let F be a p-adic field and let G be a connected reductive group defined over F. We assume p is large. Denote $\mathfrak{g}$ the Lie algebra of G. To each vertex s of the reduced Bruhat–Tits’ building of G, we associate as usual a reductive Lie algebra ${\mathfrak{g}}_{\mathit{s}}$ defined over the residual field ${\mathbb{F}}_{\mathit{q}}$. We normalize suitably a Fourier-transform $\mathit{f}\mapsto \hat{\mathit{f}}$ on ${\mathit{C}}_{\mathit{c}}^{\infty }(\mathfrak{g}(\mathit{F}))$. We study the subspace of functions $\mathit{f}\in {\mathit{C}}_{\mathit{c}}^{\infty }(\mathfrak{g}(\mathit{F}))$ such that the orbital integrals of f and of $\hat{\mathit{f}}$ are 0 for each element of $\mathfrak{g}(\mathit{F})$ which is not topologically nilpotent. This space is related to the characteristic functions of the character-sheaves on the spaces ${\mathfrak{g}}_{\mathit{s}}({\mathbb{F}}_{\mathit{q}})$, for each vertex s, which are cuspidal and with nilpotent support. We prove that our subspace behave well under endoscopy.

The goal of the paper is to explain a harmonic map approach to two geometric problems related to the Torelli map. The first is related to the existence of totally geodesic submanifolds in the image of the Torelli map, and the second is on the rigidity of representation of a lattice of a semisimple Lie group in a mapping class group.