7,889 research outputs found
Finite covers of random 3-manifolds
A 3-manifold is Haken if it contains a topologically essential surface. The
Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite
fundamental group has a finite cover which is Haken. In this paper, we study
random 3-manifolds and their finite covers in an attempt to shed light on this
difficult question. In particular, we consider random Heegaard splittings by
gluing two handlebodies by the result of a random walk in the mapping class
group of a surface. For this model of random 3-manifold, we are able to compute
the probabilities that the resulting manifolds have finite covers of particular
kinds. Our results contrast with the analogous probabilities for groups coming
from random balanced presentations, giving quantitative theorems to the effect
that 3-manifold groups have many more finite quotients than random groups. The
next natural question is whether these covers have positive betti number. For
abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show
that the probability of positive betti number is 0.
In fact, many of these questions boil down to questions about the mapping
class group. We are lead to consider the action of mapping class group of a
surface S on the set of quotients pi_1(S) -> Q. If Q is a simple group, we show
that if the genus of S is large, then this action is very mixing. In
particular, the action factors through the alternating group of each orbit.
This is analogous to Goldman's theorem that the action of the mapping class
group on the SU(2) character variety is ergodic.Comment: 60 pages; v2: minor changes. v3: minor changes; final versio
A bound on the expected number of random elements to generate a finite group all of whose Sylow subgroups are d-generated
Assume that all the Sylow subgroups of a finite group G can be generated by d elements. Then the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators is found, is at most d+ \u3b7 with \u3b7 3c 2.875065
A bound on the expected number of random elements to generate a finite group all of whose Sylow subgroups are d-generated
Assume that all the Sylow subgroups of a finite group can be generated by
elements. Then the expected number of elements of which have to be
drawn at random, with replacement, before a set of generators is found, is at
most with $\eta \sim 2.875065.
Invariable generation and the chebotarev invariant of a finite group
A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si
> for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the
minimal size of an invariable generating set for an arbitrary finite group G.
In response to a question in [KZ] we also bound the size of a randomly chosen
set of elements of G that is likely to generate G invariably. Along the way we
prove that every finite simple group is invariably generated by two elements.Comment: Improved versio
The expected number of random elements to generate a finite group
We will see that the expected number of elements of a finite group G which have to be drawn at random, with replacement, before a set of generators is found, can be determined using the M\uf6bius function defined on the subgroup lattice of G. We will discuss several applications of this result
Structure computation and discrete logarithms in finite abelian p-groups
We present a generic algorithm for computing discrete logarithms in a finite
abelian p-group H, improving the Pohlig-Hellman algorithm and its
generalization to noncyclic groups by Teske. We then give a direct method to
compute a basis for H without using a relation matrix. The problem of computing
a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian
group G is addressed, yielding a Monte Carlo algorithm to compute the structure
of G using O(|G|^0.5) group operations. These results also improve generic
algorithms for extracting pth roots in G.Comment: 23 pages, minor edit
Finite groups with large Chebotarev invariant
A subset of a finite group is said to invariably
generate if the set generates for
every choice of . The Chebotarev invariant of is the
expected value of the random variable that is minimal subject to the
requirement that randomly chosen elements of invariably generate .
The authors recently showed that for each , there exists a constant
such that . This
bound is asymptotically best possible. In this paper we prove a partial
converse: namely, for each there exists an absolute constant
such that if is a finite group and
, then has a section such that , and for some prime
power , with
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