We present a practical algorithm which, given a non-archimedean local field $K$ and any two elements $A,B\in {\rm SL_2}(K)$, determines after finitely many steps whether or not the subgroup $\langle A, B \rangle\le {\rm SL_2}(K)$ is discrete and free of rank two. This makes use of the Ping Pong Lemma applied to the action of ${\rm SL_2}(K)$ by isometries on its Bruhat-Tits tree. The algorithm itself can also be used for two-generated subgroups of the isometry group of any locally finite simplicial tree, and has applications to the constructive membership problem. In an appendix joint with Frédéric Paulin, we give an erratum to his 1989 paper `The Gromov topology on $\mathbb{R}$-trees', which details some translation length formulae that are fundamental to the algorithm.
Let $G$ be a higher rank semisimple linear algebraic group over a non-Archimedean local field. The simplicial complexes corresponding to any sequence of pairwise non-conjugate irreducible lattices in $G$ are Benjamini-Schramm convergent to the Bruhat-Tits building. Convergence of the relative Plancherel measures and normalized Betti numbers follows. This extends the work of Abert, Bergeron, Biringer, Gelander, Nokolov, Raimbault and Samet from real Lie groups to linear groups over arbitrary local fields. Along the way, various results concerning Invariant Random Subgroups and in particular a v…
Using Cayley transform, we show how to construct rotation matrices \emph{infinitely near} the identity matrix over a non-archimedean pythagorean field. As an application, an alternative way to construct non-central proper normal subgroups of the rotation group over such fields is provided.
Diffeomorphism groups $G$ of manifolds $M$ on locally $\bf F$-convex spaces over non-Archimedean fields $\bf F$ are investigated. It is shown that their structure has many differences with the diffeomorphism groups of real and complex manifolds. It is proved that $G$ is not a Banach-Lie group, but it has a neighbourhood $W$ of the unit element $e$ such that each element $g$ in $W$ belongs to at least one corresponding one-parameter subgroup. It is proved that $G$ is simple and perfect. Its compact subgroups $G_c$ are studied such that a dimension over $\bf F$ of its tangent space $dim_{\bf F}T…
Let f be a nondegenerate quadratic form in at least 5 variables over a number field K and let S be a finite set of valuations of K containing all Archimedean ones. We prove that if the Witt index of f is at least 2 or it is 1 and S contains a non-Archimedean valuation, then the S-arithmetic subgroups of the special orthogonal group of f have bounded generation. These groups provide a series of examples of boundedly generated S-arithmetic groups in isotropic, but not quasi-split, algebraic groups.
We show that connected separable locally compact groups are infinitesimally finitely generated, meaning that there is an integer $n$ such that every neighborhood of the identity contains $n$ elements generating a dense subgroup. We generalize a theorem of Schreier and Ulam by showing that any separable connected compact group is infinitesimally $2$-generated. Inspired by a result of Kechris, we introduce the notion of a quasi non-archimedean group. We observe that full groups are quasi non-archimedean, and that every continuous homomorphism from an infinitesimally finitely generated group into…
Let A be a locally compact group topologically generated by d elements and let k>d. Consider the action, by pre-composition, of Aut(F_k) on the set of marked, k-generated, dense subgroups D_{k,A} := {h:F_k --> A | h(F_k) is dense in A}. We prove the ergodicity of this action for two families of simple, totally disconnected locally compact groups. (i) A = PSL(2,K) where K is a non-Archimedean local field (of characteristic not equal to 2), (ii) A = Aut^{0}(T) - the group of orientation preserving automorphisms of a (q+1)-regular tree, for q > 1. In contrast, a recent result of Minsky's shows th…
Let F be a global field, and let S be a finite set of places of F containing all archimedean places. Consider the product X of the symmetric spaces and Bruhat-Tits buildings for PGL_d of the completions of F at archimedean and non-archimedean places in S, respectively. We construct compactifications of the quotient of X by S-arithmetic subgroups of PGL_d(F). The constructions make delicate use of reductive Borel-Serre spaces for archimedean places and polyhedral and seminorm compactifications at nonarchimedean places. We also briefly discuss a few potential applications of our compacifications…
The purpose of this paper is to link anisotropy properties of an algebraic group together with compactness issues in the topological group of its rational points. We nd equivalent conditions on a smooth ane algebraic group scheme over a non-Archimedean local eld for the associated rational points to admit maximal compact subgroups. We use the structure theory of pseudo-reductive groups provided, whatever the characteristic, by Conrad, Gabber and Prasad. We also investigate thoroughly maximal prop subgroups in the semisimple case, using Bruhat-Tits theory.
We show that the Heisenberg type group $H_X=(\Bbb{Z}_2 \oplus V) \leftthreetimes V^{\ast}$, with the discrete Boolean group $V:=C(X,\Z_2)$, canonically defined by any Stone space $X$, is always minimal. That is, $H_X$ does not admit any strictly coarser Hausdorff group topology. This leads us to the following result: for every (locally compact) non-archimedean $G$ there exists a (resp., locally compact) non-archimedean minimal group $M$ such that $G$ is a group retract of $M.$ For discrete groups $G$ the latter was proved by S. Dierolf and U. Schwanengel. We unify some old and new characteriza…
We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dense subgroups with non-empty domain of discontinuity have discrete commensurators. This generalizes a Theorem of Greenberg for Kleinian groups. We then prove that for all finitely generated, Zariski dense, infinite covolume discrete subgroups of $Isom ({\mathbb{H}}^3)$, commensurat…
We prove that a free group F_2 admits a faithful discrete representation into Diff_{+}(I). We also prove that F_2 admits a faithful discrete representation into Homeo_{+}(I). Some properties of these representations have been studied. In the last section we raise several questions.
Rational discrete cohomology and homology for a totally disconnected locally compact group $G$ is introduced and studied. The $\mathrm{Hom}$-$\otimes$ identities associated to the rational discrete bimodule $\mathrm{Bi}(G)$ allow to introduce the notion of rational duality groups in analogy to the discrete case. It is shown that semi-simple groups defined over a non-discrete, non-archimedean local field are rational t.d.l.c. duality groups, and the same is true for certain topological Kac-Moody groups. However, Y. Neretin's group of spheromorphisms of a locally finite regular tree is not even …
Among connected linear algebraic groups, quasi-reductive groups generalize pseudo-reductive groups, which in turn form a useful relaxation of the notion of reductivity. We study quasi-reductive groups over non-archimedean local fields, focusing on aspects involving their locally compact topology. For such groups we construct valuated root data (in the sense of Bruhat--Tits) and we make them act nicely on affine buildings. We prove that they admit Iwasawa and Cartan decompositions, and we construct small compact open subgroups with an Iwahori decomposition. We also initiate the smooth represent…
We initiate the study of affine actions of groups on $Λ$-trees for a general ordered abelian group $Λ$; these are actions by dilations rather than isometries. This gives a common generalisation of isometric action on a $Λ$-tree, and affine action on an $\R$-tree as studied by I. Liousse. The duality between based length functions and actions on $Λ$-trees is generalised to this setting. We are led to consider a new class of groups: those that admit a free affine action on a $Λ$-tree for some $Λ$. Examples of such groups are presented, including soluble Baumslag-Solitar groups and the discrete H…
We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show for this that the linearity of a countable Kac-Moody group implies the existence of a closed embedding of the corresponding topological group in a non-Archimedean simple Lie group, thanks to a commensurator …
We amend the statement of point~(i) in Theorem~1.3 in arxiv:0901.1022 and supply the additional arguments and minor changes for the results that depend on it. We also seize the occasion and generalize to non-finitely generated lattices.
Let $G$ be a semisimple real algebraic Lie group of real rank at least two and $U$ be the unipotent radical of a non-trivial parabolic subgroup. We prove that a discrete Zariski dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$. This solves a conjecture of Margulis and extends previous work of Hee Oh.
We prove that, if a topological group $G$ has an open subgroup of infinite index, then every net of tight Borel probability measures on $G$ UEB-converging to invariance dissipates in $G$ in the sense of Gromov. In particular, this solves a 2006 problem by Pestov: for every left-invariant (or right-invariant) metric $d$ on the infinite symmetric group $\mathrm{Sym}(\mathbb{N})$, compatible with the topology of pointwise convergence, the sequence of the finite symmetric groups $\left(\mathrm{Sym}(n),d\!\!\upharpoonright_{\mathrm{Sym}(n)},μ_{\mathrm{Sym}(n)}\right)_{n \in \mathbb{N}}$ equipped wi…
We study free topological groups defined over uniform spaces in some subclasses of the class NA of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean NA groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another contrasting advantage is that the induced topological group actions on free abelian NA group…
We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field $K$. To capture the features of classical amenability that induce the vanishing of real bounded cohomology, we introduce the notion of normed $K$-amenability, of which we prove an algebraic characterization. It implies that normed $K$-amenable groups are locally elliptic, and it relates an invariant, the norm of a $K$-amenable group, to the order of its discrete finite $p$-subquotients, where $p$ is the characteristic of the residue field of $K$. Moreover, w…
We study relatively minimal subgroups in topological groups. We find, in particular, some natural relatively minimal subgroups in unipotent groups which are defined over "good" rings. By "good" rings we mean archimedean absolute valued (not necessarily associative) division rings. Some of the classical rings which we consider besides the field of reals are the ring of quaternions and the ring of octonions. This way we generalize in part a previous result which was obtained by Dikranjan and Megrelishvili and involved the Heisenberg group.
For any discrete, torsion-free subgroup $Γ$ of $\mathrm{Sp}(n,1)$ (resp.\ $\mathrm{F}_4^{-20}$) with no parabolic elements, we prove that $H_{4n-1}(Γ;V)=0$ (resp.\ $H_i(Γ;V)=0$ for $i=13,14,15$) for any $Γ$--module $V$. The main technical advance is a new bound on the $p$--Jacobian of the barycenter map of Besson--Courtois--Gallot. We also apply this estimate to obtain an inequality between the critical exponent and homological dimension of $Γ$, improving on work of M.~Kapovich.
We prove a weaker version of Zassenhaus Lemma (also known as Margulis Lemma) for subgroups of Diff(I). We also show that a group with commutator subgroup containing a free subsemigroup does not admit a C_0-discrete faithful representation in Diff(I).
The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if $G=N\times A$ is a connected, simply connected, nilpotent Lie group with an Abelian factor $A$, then every uniform subgroup of $G$ is the direct product of a uniform subgroup of $N$ and ${\mathbb Z}^r$ where $r=\dim A$.
In the case where $G=$SL$_{2}(F)$ for a non-archimedean local field $F$ and $Γ$ is a discrete torsion-free cocompact subgroup of $G$, there is a known relationship between the Ihara zeta function for the quotient of the Bruhat-Tits tree of $G$ by the action of $Γ$, and an alternating product of determinants of twisted Poincaré series for parabolic subgroups of the affine Weyl group of $G$. We show how this can be generalised to other split simple algebraic groups of rank two over $F$, and formulate a conjecture about how this might be generalised to groups of higher rank.
We prove that the Harish-Chandra--Schwartz space associated with a discrete subgroup of a semisimple Lie group is a dense subalgebra of the reduced $C^*$-algebra of the discrete subgroup. Then, we prove that for the reduced $C^*$-norm is controlled by the norm of the Harish-Chandra--Schwartz space. This inequality is weaker than property RD and holds for any discrete group acting isometrically, properly on a Riemannian symmetric space.
Let G be a simple non-compact linear connected Lie group and H be a closed non-compact semisimple subgroup. We are interested in finding classes of homogeneous spaces G/H admitting proper actions of discrete non virtually abelian subgroups of G. We develop an algorithm for finding such homogeneous spaces. As a testing example we obtain a list of all non-compact homogeneous spaces G/H admitting proper action of a discrete and non virtually abelian subgroup of G in the case when G has rank at most 8, and H is a maximal proper semisimple subgroup.
Let $G$ be a discrete group generated by reflections in hyperbolic or Euclidean space, and $H\subset G$ be a finite index subgroup generated by reflections. Suppose that the fundamental chamber of $G$ is a finite volume polytope with $k$ facets. In this paper, we prove that the fundamental chamber of $H$ has at least $k$ facets.
Groups acting freely on Z^n-trees (Z^n-free groups) play a key role in the study of non-archimedean group actions. Following Stallings' ideas, we develop graph-theoretic techniques to investigate subgroup structure of Z^n-free groups. As an immediate application of the presented method, we give an effective solution to the Uniform Membership Problem and the Power Problem in Z^n-free groups.
In this paper we give an alternative proof of Schreiber's theorem which says that an infinite discrete approximate subgroup in $\mathbb{R}^d$ is relatively dense around a subspace. We also deduce from Schreiber's theorem two new results. The first one says that any infinite discrete approximate subgroup in $\mathbb{R}^d$ is a restriction of a Meyer set to a thickening of a linear subspace in $\mathbb{R}^d$, and the second one provides an extension of Schreiber's theorem to the case of the Heisenberg group.
We show that if $G$ is a non-archimedean, Roelcke precompact, Polish group, then $G$ has Kazhdan's property (T). Moreover, if $G$ has a smallest open subgroup of finite index, then $G$ has a finite Kazhdan set. Examples of such $G$ include automorphism groups of countable $ω$-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation groups. The latter allows the construction of a non-abelian free subgroup of $G…
Let G be the group GL(N,F), where F is a non-archimedean locally compact field. Using Bushnell and Kutzko's simple types, as well as an original idea of Henniart's, we construct explicit pseudo-coefficients for the discrete series representations of G. As an application we deduce new formulas for the value of the Harish-Chandra character of certain such representations at certain elliptic regular elements.
Dehornoy showed that the Artin braid groups $B_n$ are left-orderable. This ordering is discrete, but we show that, for $n >2$ the Dehornoy ordering, when restricted to certain natural subgroups, becomes a dense ordering. Among subgroups which arise are the commutator subgroup and the kernel of the Burau representation (for those $n$ for which the kernel is nontrivial). These results follow from a characterization of least positive elements of any normal subgroup of $B_n$ which is discretely ordered by the Dehornoy ordering.
