34 research outputs found
Bertrand's postulate and subgroup growth
In this article we investigate the L^1-norm of certain functions on groups
called divisibility functions. Using these functions, their connection to
residual finiteness, and integration theory on profinite groups, we define the
residual average of a finitely generated group. One of the main results in this
article is the finiteness of residual averages on finitely generated linear
groups. Whether or not the residual average is finite depends on growth rates
of indices of finite index subgroups. Our results on index growth rates are
analogous to results on gaps between primes, and provide a variant of the
subgroup growth function, which may be of independent interest.Comment: 33 page
Asymptotic growth and least common multiples in groups
In this article we relate word and subgroup growth to certain functions that
arise in the quantification of residual finiteness. One consequence of this
endeavor is a pair of results that equate the nilpotency of a finitely
generated group with the asymptotic behavior of these functions. The second
half of this article investigates the asymptotic behavior of two of these
functions. Our main result in this arena resolves a question of Bogopolski from
the Kourovka notebook concerning lower bounds of one of these functions for
nonabelian free groups.Comment: 13 page
Residual Finiteness Growths of Virtually Special Groups
Let be a virtually special group. Then the residual finiteness growth of
is at most linear. This result cannot be found by embedding into a
special linear group. Indeed, the special linear group
, for , has residual finiteness growth
.Comment: Updated version contains minor changes incorporating referee
comments/suggestions and a simplified proof of Lemma 4.
Minsky machines and algorithmic problems
This is a survey of using Minsky machines to study algorithmic problems in
semigroups, groups and other algebraic systems.Comment: 19 page
Stationary distributions for diffusions with inert drift
Consider reflecting Brownian motion in a bounded domain in that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting Brownian motion and the value of the drift vector has a product form. Moreover, the first component is uniformly distributed on the domain, and the second component has a Gaussian distribution. We also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential