The subnormal structure of classical-like groups over commutative rings

Let 𝑛 be an integer greater than or equal to 3, and let (R,\Delta) be a Hermitian form ring, where 𝑅 is commutative. We prove that if 𝐻 is a subgroup of the odd-dimensional unitary group \operatorname{U}_{2n+1}(R,\Delta) normalised by a relative elementary subgroup \operatorname{EU}_{2n+1}((R,\Delta),(I,\Omega)), then there is an odd form ideal (J,\Sigma) such that\operatorname{EU}_{2n+1}((R,\Delta),(JI^{k},\Omega_{\mathrm{min}}^{JI^{k}}\dotplus\Sigma\circ I^{k}))\leq H\leq\operatorname{CU}_{2n+1}((R,\Delta),(J,\Sigma)),where k=12 if n=3 respectively k=10 if n\geq 4. As a consequence of this result, we obtain a sandwich theorem for subnormal subgroups of odd-dimensional unitary groups.

Generation of relative commutator subgroups in Chevalley groups. II

In the present paper, which is a direct sequel of our paper [14] joint with Roozbeh Hazrat, we prove an unrelativized version of the standard commutator formula in the setting of Chevalley groups. Namely, let Φ be a reduced irreducible root system of rank ≥ 2, let R be a commutative ring and let I,J be two ideals of R. We consider subgroups of the Chevalley group G(Φ, R) of type Φ over R. The unrelativized elementary subgroup E(Φ, I) of level I is generated (as a group) by the elementary unipotents xα(ξ), α ∈ Φ, ξ ∈ I, of level I. Obviously, in general, E(Φ, I) has no chance to be normal in E(Φ, R); its normal closure in the absolute elementary subgroup E(Φ, R) is denoted by E(Φ, R, I). The main results of [14] implied that the commutator [E(Φ, I), E(Φ, J)] is in fact normal in E(Φ, R). In the present paper we prove an unexpected result, that in fact [E(Φ, I), E(Φ, J)] = [E(Φ, R, I), E(Φ, R, J)]. It follows that the standard commutator formula also holds in the unrelativized form, namely [E(Φ, I), C(Φ, R, J)] = [E(Φ, I), E(Φ, J)], where C(Φ, R, I) is the full congruence subgroup of level I. In particular, E(Φ, I) is normal in C(Φ, R, I).

Injective stability for odd-dimensional unitary K1{K_{1}}

In this paper, under the usual stable range condition, a decomposition theorem for the elementary subgroup is obtained, and the injective stability theorem for odd-dimensional unitary {K_{1}} is proved.

Recurrence of quadratic differentials for harmonic measure

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmüller geodesic typical with respect to the harmonic measure for such random walks, is recurrent to the thick part of the principal stratum. As a consequence, the vertical foliation of such a random Teichmüller geodesic has no saddle connections.

Contracting elements and random walks

We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper {\mathrm{CAT}(0)} spaces, elements acting hyperbolically on the Bass–Serre tree in graph manifold groups. We also define a related notion of weakly contracting element, and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in {\mathrm{Out}(F_{n})} , {n\geq 3} . We show that each weakly contracting element is contained in a hyperbolically embedded elementary subgroup, which allows us to answer a problem in [16]. We prove that any simple random walk in a non-elementary finitely generated subgroup containing a (weakly) contracting element ends up in a non-(weakly-)contracting element with exponentially decaying probability.

The stratum of random mapping classes

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anosov, with invariant Teichmüller geodesics in the principal stratum. This provides an answer to a question of Kapovich and Pfaff [Internat. J. Algebra Comput.25, 2015 (5) 745–798].

Structure of Hyperbolic Unitary Groups II: Classification of E-Normal Subgroups

This paper proves the sandwich classification conjecture for subgroups of an even dimensional hyperbolic unitary group [Formula: see text] which are normalized by the elementary subgroup [Formula: see text], under the condition that R is a quasi-finite ring with involution, i.e., a direct limit of module finite rings with involution, and [Formula: see text].

Generation of relative commutator subgroups in Chevalley groups

Let Φ be a reduced irreducible root system of rank greater than or equal to 2, let R be a commutative ring and let I, J be two ideals of R. In the present paper we describe generators of the commutator groups of relative elementary subgroups [E(Φ,R,I),E(Φ,R,J)] both as normal subgroups of the elementary Chevalley group E(Φ,R), and as groups. Namely, let xα(ξ), α ∈ Φ ξ ∈ R, be an elementary generator of E(Φ,R). As a normal subgroup of the absolute elementary group E(Φ,R), the relative elementary subgroup is generated by xα(ξ), α ∈ Φ, ξ ∈ I. Classical results due to Stein, Tits and Vaserstein assert that as a group E(Φ,R,I) is generated by zα(ξ,η), where α ∈ Φ, ξ ∈ I, η ∈ R. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of E(Φ,R) the relative commutator subgroup [E(Φ,R,I),E(Φ,R,J)] is generated by the following three types of generators: (i) [xα(ξ),zα(ζ,η)], (ii) [xα(ξ),x_α(ζ)] and (iii) xα(ξζ), where α ∈ Φ, ξ ∈ I, ζ ∈ J, η ∈ R. As a group, the generators are essentially the same, only that type (iii) should be enlarged to (iv) zα(ξζ,η). For classical groups, these results, with many more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results in the recent work of Stepanov on relative commutator width.