arXiv subjects

Dave Witte

Publications and source records attributed to Dave Witte.

At least 19 recordsLinked to original sources

arXiv2009

Transitive permutation groups of prime-squared degree

We explicitly determine all of the transitive groups of degree p-squared, p a prime, whose Sylow p-subgroup is not the wreath product of two cyclic groups of order p. Furthermore, we provide a general description of the transitive groups of degree p-squared whose Sylow p-subgroup is such a wreath product, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p-squared, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p-squared, and find all nonnormal Cayley graphs of order p-squared.

math.GR↗

arXiv2002

Real representations of semisimple Lie algebras have Q-forms

We prove that each real semisimple Lie algebra G has a Q-form, such that every real representation of G can be realized over the rational numbers Q. This was previously proved by M.S.Raghunathan (and rediscovered by P.Eberlein) in the special case where G is compact.

math.RT↗

arXiv2002

Ergodic actions of semisimple Lie groups on compact principal bundles

Let G = SL(n,R) (or, more generally, let G be a connected, noncompact, simple Lie group). For any compact Lie group K, it is easy to find a compact manifold M, such that there is a volume-preserving, connection-preserving, ergodic action of G on some smooth, principal K-bundle P over M. Can M can be chosen independent of K? We show that if M = H/L is a homogeneous space, and the action of G on M is by translations, then P must also be a homogeneous space H'/L'. Consequently, there is a strong restriction on the groups K that can arise over this particular M.

math.DG↗

arXiv2001

Superrigid subgroups and syndetic hulls in solvable Lie groups

This is an expository paper. It is not difficult to see that every group homomorphism from the additive group Z of integers to the additive group R of real numbers extends to a homomorphism from R to R. We discuss other examples of discrete subgroups D of connected Lie groups G, such that the homomorphisms defined on D can ("virtually") be extended to homomorphisms defined on all of G. For the case where G is solvable, we give a simple proof that D has this property if it is Zariski dense. The key ingredient is a result on the existence of syndetic hulls.

math.RT↗

arXiv2001

Hamiltonian Paths in Cartesian Powers of Directed Cycles

The vertex set of the kth cartesian power of a directed cycle of length m can be naturally identified with the set of k-tuples of integers modulo m. For any two vertices v and w of this graph, it is easy to see that if there is a hamiltonian path from v to w, then the sum of the coordinates of v is congruent, modulo m, to one more than the sum of the coordinates of w. We prove the converse, unless k = 2 and m is odd.

math.CO↗

arXiv2001

Tessellations of homogeneous spaces of classical groups of real rank two

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a "tessellation" if there is a discrete subgroup D of G, such that D acts properly discontinuously on G/H, and the double-coset space D\G/H is compact. Note that if either H or G/H is compact, then G/H has a tessellation; these are the obvious examples. It is not difficult to see that if G has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when G has real rank two. In particular, R.Kulkarni and T.Kobayashi constructed examples that are not obvious when G = SO(2,2n) or SU(2,2n). H.Oh and D.Witte constructed additional examples in both of these cases, and obtained a complete classification when G = SO(2,2n). We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G = SU(2,2n). This includes the construction of another family of examples. The main results are obtained from methods of Y.Benoist and T.Kobayashi: we fix a Cartan decomposition G = KAK, and study the intersection of KHK with A. Our exposition generally assumes only the standard theory of connected Lie groups, although basic properties of real algebraic groups are sometimes also employed; the specialized techniques that we use are developed from a fairly elementary level.

math.RT↗

arXiv2000

Groups that do not act by automorphisms of codimension-one foliations

Let G be a finitely generated group having the property that any action of any finite-index subgroup of G by homeomorphisms of the circle must have a finite orbit. (By a theorem of E.Ghys, lattices in simple Lie groups of real rank at least two have this property.) Suppose that such a G acts on a compact manifold M by automorphisms of a codimension-one C2 foliation, F. We show that if F has a compact leaf, then some finite-index subgroup of G fixes a compact leaf of F. Furthermore, we give sufficient conditions for some finite-index subgroup of G to fix each leaf of F.

math.GT↗

arXiv2000

Homogeneous Lorentz manifolds with simple isometry group

Let H be a closed, noncompact subgroup of a simple Lie group G, such that G/H admits an invariant Lorentz metric. We show that if G = SO(2,n), with n > 2, then the identity component of H is conjugate to the identity component of SO(1,n). Also, if G = SO(1,n), with n > 2, then the identity component of H is conjugate to the identity component of SO(1,n-1).

math.DG↗

arXiv2000

On automorphisms of arithmetic subgroups of unipotent groups in positive characteristic

Let F be a local field of positive characteristic, and let G be either a Heisenberg group over F, or a certain (nonabelian) two-dimensional unipotent group over F. If H is an arithmetic subgroup of G, we provide an explicit description of every automorphism of H. From this description, it follows that every automorphism of H virtually extends to a virtual automorphism of G.

math.GR↗

arXiv2000

Cartan-decomposition subgroups of SU(2,n)

We give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G = SU(2,n) has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan decomposition G = K A K of G, and then carry out an approximate calculation of the intersection of KHK with A, for each closed, connected subgroup H of G. This generalizes the work of Hee Oh and Dave Witte for G = SO(2,n).

math.RT↗

arXiv2000

Actions of semisimple Lie groups on circle bundles

Suppose G is a connected, simple, real Lie group with real rank at least two, M is an ergodic G-space with invariant probability measure, and f is a Homeo(T)-valued Borel cocycle, where Homeo(T) denotes the group of homeomorphisms of the circle T. We use an argument of E.Ghys to show that there is a G-invariant probability measure on the skew product of M and T. Furthermore, if the image of f consists of diffeomorphisms, then there is an invariant measure that is equivalent to the product measure; therefore, f is cohomologous to a cocycle with values in the isometry group of T.

math.DS↗

arXiv1999

Cartan-decomposition subgroups of SO(2,n)

For G = SL(3,R) and G = SO(2,n), we give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan decomposition G = KAK of G, and then carry out an approximate calculation of the intersection of KHK with A, for each closed, connected subgroup H of G.

math.RT↗

arXiv1999

Compact Clifford-Klein forms of homogeneous spaces of SO(2,n)

A homogeneous space G/H is said to have a compact Clifford-Klein form if there exists a discrete subgroup D of G that acts properly discontinuously on G/H, such that the quotient space D\G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2,n), such that G/H has a compact Clifford-Klein form, but our classification is not quite complete when n is odd. The work reveals new examples of homogeneous spaces of SO(2,n) that have compact Clifford-Klein forms, if n is even. Furthermore, we show that if H is a closed, connected subgroup of G = SL(3,R), and neither H nor G/H is compact, then G/H does not have a compact Clifford-Klein form, and we also study noncompact Clifford-Klein forms of finite volume.

math.RT↗

arXiv1998

Foliation-preserving Maps Between Solvmanifolds

For i = 1,2, let Gamma_i be a lattice in a simply connected, solvable Lie group G_i, and let X_i be a connected Lie subgroup of G_i. The double cosets Gamma_igX_i provide a foliation F_i of the homogeneous space Gamma_i\G_i. Let f be a continuous map from Gamma_1\G_1 to Gamma_2\G_2 whose restriction to each leaf of F_1 is a covering map onto a leaf of F_2. If we assume that F_1 has a dense leaf, and make certain technical technical assumptions on the lattices Gamma_1 and Gamma_2, then we show that f must be a composition of maps of two basic types: a homeomorphism of Gamma_1\M_1 that takes each leaf of F_1 to itself, and a map that results from twisting an affine map by a homomorphism into a compact group. We also prove a similar result for many cases where G_1 and G_2 are neither solvable nor semisimple.

math.GT↗

arXiv1997

Automorphism groups with cyclic commutator subgroup and Hamilton cycles

It has been shown that there is a Hamilton cycle in every connected Cayley graph on each group G whose commutator subgroup is cyclic of prime-power order. This paper considers connected, vertex-transitive graphs X of order at least 3 where the automorphism group of X contains a transitive subgroup G whose commutator subgroup is cyclic of prime-power order. We show that of these graphs, only the Petersen graph is not hamiltonian.

math.CO↗

arXiv1996

Archimedean superrigidity of solvable S-arithmetic groups

Let $\Ga$ be a connected, solvable linear algebraic group over a number field~$K$, let $S$ be a finite set of places of~$K$ that contains all the infinite places, and let $\theints$ be the ring of $S$-integers of~$K$. We define a certain closed subgroup~$\GOS$ of $\Ga_S = \prod_{v \in S} \Ga_{K_v}$ that contains $\Ga_{\theints}$, and prove that $\Ga_{\theints}$ is a superrigid lattice in~$\GOS$, by which we mean that finite-dimensional representations $\alpha\colon \Ga_{\theints} \to \GL_n(\real)$ more-or-less extend to representations of~$\GOS$. The subgroup~$\GOS$ may be a proper subgroup of~$\Ga_S$ for only two reasons. First, it is well known that $\Ga_{\theints}$ is not a lattice in~$\Ga_S$ if $\Ga$ has nontrivial $K$-characters, so one passes to a certain subgroup $\GS$. Second, $\Ga_{\theints}$ may fail to be Zariski dense in $\GS$ in an appropriate sense; in this sense, the subgroup $\GOS$ is the Zariski closure of~$\Ga_{\theints}$ in~$\GS$. Furthermore, we note that a superrigidity theorem for many non-solvable $S$-arithmetic groups can be proved by combining our main theorem with the Margulis Superrigidity Theorem.

math.RT↗

arXiv1996

Prehomogeneous vector spaces and ergodic theory II

We apply M. Ratner's theorem on closures of unipotent orbits to the study of three families of prehomogeneous vector spaces. As a result, we prove analogues of the Oppenheim Conjecture for simultaneous approximation by values of certain alternating bilinear forms in an even number of variables and certain alternating trilinear forms in six and seven variables.

math.RT↗

arXiv1996

Cocycle superrigidity for ergodic actions of non-semisimple Lie groups

Suppose $L$ is a semisimple Levi subgroup of a connected Lie group~$G$, $X$ is a Borel $G$-space with finite invariant measure, and $\alpha \colon X \times G \to \GL_n(\real)$ is a Borel cocycle. Assume $L$ has finite center, and that the real rank of every simple factor of~$L$ is at least two. We show that if $L$ is ergodic on~$X$, and the restriction of~$\alpha$ to~$X \times L$ is cohomologous to a homomorphism (modulo a compact group), then, after passing to a finite cover of~$X$, the cocycle $\alpha$ itself is cohomologous to a homomorphism (modulo a compact group).

math.RT↗