175 research outputs found
On the number and location of short geodesics in moduli space
A closed Teichmuller geodesic in the moduli space M_g of Riemann surfaces of
genus g is called L-short if it has length at most L/g. We show that, for any L
> 0, there exist e_2 > e_1 > 0, independent of g, so that the L-short geodesics
in M_g all lie in the intersection of the e_1-thick part and the e_2-thin part.
We also estimate the number of L-short geodesics in M_g, bounding this from
above and below by polynomials in g whose degrees depend on L and tend to
infinity as L does.Comment: 23 pages, 1 figur
Finite rigid sets in curve complexes
We prove that curve complexes of surfaces are finitely rigid: for every
orientable surface S of finite topological type, we identify a finite
subcomplex X of the curve complex C(S) such that every locally injective
simplicial map from X into C(S) is the restriction of an element of Aut(C(S)),
unique up to the (finite) point-wise stabilizer of X in Aut(C(S)). Furthermore,
if S is not a twice-punctured torus, then we can replace Aut(C(S)) in this
statement with the extended mapping class group.Comment: 19 pages, 12 figures. v2: small additions to improve exposition. v3:
conclusion of Lemma 2.5 weakened, and proof of Theorem 3.1 adjusted
accordingly. Main theorem remains unchange
Abstract commensurators of braid groups
Let B_n be the braid group on n strands, with n at least 4, and let Mod(S) be
the extended mapping class group of the sphere with n+1 punctures. We show that
the abstract commensurator of B_n is isomorphic to a semidirect product of
Mod(S) with a group we refer to as the transvection subgroup, Tv(B_n). We also
show that Tv(B_n) is itself isomorphic to a semidirect product of an infinite
dimensional rational vector space with the multiplicative group of nonzero
rational numbers.Comment: 10 page
Connectivity of the space of ending laminations
We prove that for any closed surface of genus at least four, and any punctured surface
of genus at least two, the space of ending laminations is connected. A theorem of E.
Klarreich [28, Theorem 1.3] implies that this space is homeomorphic to the Gromov
boundary of the complex of curves. It follows that the boundary of the complex of curves
is connected in these cases, answering the conjecture of P. Storm. Other applications
include the rigidity of the complex of curves and connectivity of spaces of degenerate
Kleinian groups
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