Volume 72 Issue none | Michigan Mathematical Journal (original) (raw)

Yves Benoist, Jean-François Quint

Michigan Math. J. 72, 3-23, (August 2022) DOI: 10.1307/mmj/20217205

KEYWORDS: 22E20, 19C09

We show that, in a weakly regular _p_-adic Lie group G, the subgroup Gu 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 Gu

M. R. Bridson, A. W. Reid

Michigan Math. J. 72, 25-49, (August 2022) DOI: 10.1307/mmj/20217218

KEYWORDS: 20H10, 20E18

Let Γ be a nonelementary Kleinian group and H<Γ 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 Vol(3) and that of the 4-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in PSL(2,C) is profinitely rigid, then so is its normalizer in PSL(2,C).

Michigan Math. J. 72, 51-76, (August 2022) DOI: 10.1307/mmj/20217214

KEYWORDS: 14L15, 14D20, 14K05, 14L30, 20G15

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

Jean-Louis Colliot-Thélène, David Harbater, Julia Hartmann, Daniel Krashen, R. Parimala, V. Suresh

Michigan Math. J. 72, 77-144, (August 2022) DOI: 10.1307/mmj/20217219

KEYWORDS: 11E72, 12G05, 14G05, 14H25, 20G15, 14G27

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.

Henri Darmon, Michael Harris, Victor Rotger, Akshay Venkatesh

Michigan Math. J. 72, 145-207, (August 2022) DOI: 10.1307/mmj/20217221

KEYWORDS: 11G18, 14G35

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.

Michigan Math. J. 72, 209-242, (August 2022) DOI: 10.1307/mmj/20217211

KEYWORDS: 20G15, 11E72, 14J27, 14J28

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.

Sam Edwards, Minju Lee, Hee Oh

Michigan Math. J. 72, 243-259, (August 2022) DOI: 10.1307/mmj/20217222

KEYWORDS: 22E40, 37A17

In the late seventies, Sullivan showed that, for a convex cocompact subgroup Γ of SO∘(n,1) with critical exponent δ>0, any Γ-conformal measure on ∂Hn 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 Γ∖G including Haar measures.

Michigan Math. J. 72, 261-330, (August 2022) DOI: 10.1307/mmj/20217210

KEYWORDS: 20G25, 11E72, 20E18, 22E41

Let F be a _p_-adic field, that is, a finite extension of Qp. Let D be a finite dimensional division algebra over F, and let SL1(D) be the group of elements of reduced norm 1 in D. Prasad and Raghunathan proved that H2(SL1(D),R/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 Q2. In this paper we give an explicit upper bound for the order of H2(SL1(D),R/Z) for p≥5 and determine H2(SL1(D),R/Z) precisely when F is cyclotomic, p≥19, and the degree of D is not a power of p.

Michigan Math. J. 72, 331-342, (August 2022) DOI: 10.1307/mmj/20217217

KEYWORDS: 22E50, 20C20, 20G05, 20G25

Let F be a nonarchimedean local field of residual characteristic p≠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.

Skip Garibaldi, Robert M. Guralnick

Michigan Math. J. 72, 343-387, (August 2022) DOI: 10.1307/mmj/20217216

KEYWORDS: 20G05, 14L24

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 G(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 dimV/G<dimG, or there is a v∈V whose stabilizer in G is trivial.

T. Gelander, A. Levit, G. A. Margulis

Michigan Math. J. 72, 389-438, (August 2022) DOI: 10.1307/mmj/20217209

KEYWORDS: 22E40, 57S30, 60G10, 22F30, 22E46

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 βrδ for some explicit constants β,δ>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.

Michigan Math. J. 72, 439-448, (August 2022) DOI: 10.1307/mmj/20217208

KEYWORDS: 14L15, 20G35

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.

Michigan Math. J. 72, 449-463, (August 2022) DOI: 10.1307/mmj/20207202

KEYWORDS: 22E50, 11F70

No abstract available

Alexander Lubotzky, Raz Slutsky

Michigan Math. J. 72, 465-477, (August 2022) DOI: 10.1307/mmj/20217204

KEYWORDS: 22E40, 20G30

Abert, Gelander, and Nikolov [AGR17] conjectured that the number of generators d(Γ) of a lattice Γ in a high rank simple Lie group H grows sublinearly with v=μ(H/Γ), the co-volume of Γ in H. We prove this for nonuniform lattices in a very strong form, showing that for 2-generic such _H_s, d(Γ)=OH(logv/loglogv), 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 d(Γ)=O(logv).

Keivan Mallahi-Karai, Amir Mohammadi, Alireza Salehi Golsefidy

Michigan Math. J. 72, 479-527, (August 2022) DOI: 10.1307/mmj/20217213

KEYWORDS: 22C05, 60B15, 43A77

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.

Michigan Math. J. 72, 529-541, (August 2022) DOI: 10.1307/mmj/20207201

KEYWORDS: 20G15

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

Bertrand Rémy, Amaury Thuillier, Annette Werner

Michigan Math. J. 72, 543-557, (August 2022) DOI: 10.1307/mmj/20217220

KEYWORDS: 20E42, 51E24, 14L15, 14G22

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.

Matthew Stover, Domingo Toledo

Michigan Math. J. 72, 559-597, (August 2022) DOI: 10.1307/mmj/20217215

KEYWORDS: 14F35, 20E26, 22E40

This paper studies residual finiteness of lattices in the universal cover of PU(2,1) and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in PU(2,1) or a finite covering of it. First, we prove that certain lattices in the universal cover of 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 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 PU(2,1).

Michigan Math. J. 72, 599-619, (August 2022) DOI: 10.1307/mmj/20217212

KEYWORDS: 20H05, 22E40

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

Michigan Math. J. 72, 621-641, (August 2022) DOI: 10.1307/mmj/20207203

KEYWORDS: 22E35, 22E50

Let F be a _p_-adic field and let G be a connected reductive group defined over F. We assume p is large. Denote 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 gs defined over the residual field Fq. We normalize suitably a Fourier-transform f↦fˆ on Cc∞(g(F)). We study the subspace of functions f∈Cc∞(g(F)) such that the orbital integrals of f and of fˆ are 0 for each element of g(F) which is not topologically nilpotent. This space is related to the characteristic functions of the character-sheaves on the spaces gs(Fq), for each vertex s, which are cuspidal and with nilpotent support. We prove that our subspace behave well under endoscopy.

Michigan Math. J. 72, 643-654, (August 2022) DOI: 10.1307/mmj/20217207

KEYWORDS: 53C24, 14H15, 14K10, 53C43

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.