20,515 research outputs found
The Strong Symmetric Genus Spectrum of Abelian Groups
The strong symmetric genus of a group G is the minimum genus of any compact surface on which G acts faithfully while preserving orientation. We investigate the set of positive integers which occur as the strong symmetric genus of a finite abelian group. This is called the strong symmetric genus spectrum. We prove that there are an infinite number of gaps in the strong symmetric genus spectrum of finite abelian groups. We also determine an upper bound for the size of a finite abelian group that can act faithfully on a surface of a particular genus and then find the genus of abelian groups in particular families. These formulas produce a lower bound for the density of the strong symmetric genus spectrum
Regular dessins with a given automorphism group
Dessins d'enfants are combinatorial structures on compact Riemann surfaces
defined over algebraic number fields, and regular dessins are the most
symmetric of them. If G is a finite group, there are only finitely many regular
dessins with automorphism group G. It is shown how to enumerate them, how to
represent them all as quotients of a single regular dessin U(G), and how
certain hypermap operations act on them. For example, if G is a cyclic group of
order n then U(G) is a map on the Fermat curve of degree n and genus
(n-1)(n-2)/2. On the other hand, if G=A_5 then U(G) has genus
274218830047232000000000000000001. For other non-abelian finite simple groups,
the genus is much larger.Comment: 19 page
On knot Floer width and Turaev genus
To each knot one can associated its knot Floer homology
, a finitely generated bigraded abelian group. In general, the
nonzero ranks of these homology groups lie on a finite number of slope one
lines with respect to the bigrading. The width of the homology is, in essence,
the largest horizontal distance between two such lines. Also, for each diagram
of there is an associated Turaev surface, and the Turaev genus is the
minimum genus of all Turaev surfaces for . We show that the width of knot
Floer homology is bounded by Turaev genus plus one. Skein relations for genus
of the Turaev surface and width of a complex that generates knot Floer homology
are given.Comment: 15 pages, 15 figure
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
Limit groups, positive-genus towers and measure equivalence
By definition, an -residually free tower is positive-genus if all
surfaces used in its construction are of positive genus. We prove that every
limit group is virtually a subgroup of a positive-genus -residually
free tower. By combining this with results of Gaboriau, we prove that
elementarily free groups are measure equivalent to free groups.Comment: 10 pages; no figures. Minor changes; now to appear in Ergod. Th. &
Dynam. Sy
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