Measuring sets in infinite groups

We are now witnessing a rapid growth of a new part of group theory which has become known as "statistical group theory". A typical result in this area would say something like ``a random element (or a tuple of elements) of a group G has a property P with probability p". The validity of a statement like that does, of course, heavily depend on how one defines probability on groups, or, equivalently, how one measures sets in a group (in particular, in a free group). We hope that new approaches to defining probabilities on groups outlined in this paper create, among other things, an appropriate framework for the study of the "average case" complexity of algorithms on groups.Comment: 22 page

Lagrangian Matroids: Representations of Type BnB_n

We introduce the concept of orientation for Lagrangian matroids represented in the flag variety of maximal isotropic subspaces of dimension N in the real vector space of dimension 2N+1. The paper continues the study started in math.CO/0209100.Comment: Requires amssymb.sty; 17 page

Lagrangian Pairs and Lagrangian Orthogonal Matroids

Represented Coxeter matroids of types $C_n$ and $D_n$, that is, symplectic and orthogonal matroids arising from totally isotropic subspaces of symplectic or (even-dimensional) orthogonal spaces, may also be represented in buildings of type $C_n$ and $D_n$, respectively. Indeed, the particular buildings involved are those arising from the flags or oriflammes, respectively, of totally isotropic subspaces. There are also buildings of type $B_n$ arising from flags of totally isotropic subspaces in odd-dimensional orthogonal space. Coxeter matroids of type $B_n$ are the same as those of type $C_n$ (since they depend only upon the reflection group, not the root system). However, buildings of type $B_n$ are distinct from those of the other types. The matroids representable in odd dimensional orthogonal space (and therefore in the building of type $B_n$) turn out to be a special case of symplectic (flag) matroids, those whose top component, or Lagrangian matroid, is a union of two Lagrangian orthogonal matroids. These two matroids are called a Lagrangian pair, and they are the combinatorial manifestation of the ``fork'' at the top of an oriflamme (or of the fork at the end of the Coxeter diagram of $D_n$). Here we give a number of equivalent characterizations of Lagrangian pairs, and prove some rather strong properties of them.Comment: Requires amssymb.sty; 12 pages, 2 LaTeX figure