86 research outputs found
3-manifold groups are virtually residually p
Given a prime , a group is called residually if the intersection of
its -power index normal subgroups is trivial. A group is called virtually
residually if it has a finite index subgroup which is residually . It is
well-known that finitely generated linear groups over fields of characteristic
zero are virtually residually for all but finitely many . In particular,
fundamental groups of hyperbolic 3-manifolds are virtually residually . It
is also well-known that fundamental groups of 3-manifolds are residually
finite. In this paper we prove a common generalization of these results: every
3-manifold group is virtually residually for all but finitely many .
This gives evidence for the conjecture (Thurston) that fundamental groups of
3-manifolds are linear groups
Markov chains, -trivial monoids and representation theory
We develop a general theory of Markov chains realizable as random walks on
-trivial monoids. It provides explicit and simple formulas for the
eigenvalues of the transition matrix, for multiplicities of the eigenvalues via
M\"obius inversion along a lattice, a condition for diagonalizability of the
transition matrix and some techniques for bounding the mixing time. In
addition, we discuss several examples, such as Toom-Tsetlin models, an exchange
walk for finite Coxeter groups, as well as examples previously studied by the
authors, such as nonabelian sandpile models and the promotion Markov chain on
posets. Many of these examples can be viewed as random walks on quotients of
free tree monoids, a new class of monoids whose combinatorics we develop.Comment: Dedicated to Stuart Margolis on the occasion of his sixtieth
birthday; 71 pages; final version to appear in IJA
Conference Program
Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
Recommended from our members
Topology of Arrangements and Representation Stability
The workshop “Topology of arrangements and representation stability” brought together two directions of research: the topology and geometry of hyperplane, toric and elliptic arrangements, and the homological and representation stability of configuration spaces and related families of spaces and discrete groups. The participants were mathematicians working at the interface between several very active areas of research in topology, geometry, algebra, representation theory, and combinatorics. The workshop provided a thorough overview of current developments, highlighted significant progress in the field, and fostered an increasing amount of interaction between specialists in areas of research
Recommended from our members
Geometry and Arithmetic around Hypergeometric Functions
[no abstract available
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