5,903 research outputs found
Small intersection numbers in the curve graph
Let denote the genus orientable surface with
punctures, and let . We prove the existence of infinitely
long geodesic rays in the curve graph
satisfying the following optimal intersection property: for any natural number
, the endpoints of any length subsegment intersect
times. By combining this with work of the first author, we
answer a question of Dan Margalit.Comment: 13 pages, 6 figure
Products of Farey graphs are totally geodesic in the pants graph
We show that for a surface S, the subgraph of the pants graph determined by
fixing a collection of curves that cut S into pairs of pants, once-punctured
tori, and four-times-punctured spheres is totally geodesic. The main theorem
resolves a special case of a conjecture made by Aramayona, Parlier, and
Shackleton and has the implication that an embedded product of Farey graphs in
any pants graph is totally geodesic. In addition, we show that a pants graph
contains a convex n-flat if and only if it contains an n-quasi-flat.Comment: v2: 25 pages, 16 figures. Completely rewritten, several figures added
for clarit
Convex cocompactness in mapping class groups via quasiconvexity in right-angled Artin groups
We characterize convex cocompact subgroups of mapping class groups that arise
as subgroups of specially embedded right-angled Artin groups. That is, if the
right-angled Artin group G in Mod(S) satisfies certain conditions that imply G
is quasi-isometrically embedded in Mod(S), then a purely pseudo-Anosov subgroup
H of G is convex cocompact in Mod(S) if and only if it is combinatorially
quasiconvex in G. We use this criterion to construct convex cocompact subgroups
of Mod(S) whose orbit maps into the curve complex have small Lipschitz
constants.Comment: 30 pages, 4 figure
Convex cocompactness and stability in mapping class groups
We introduce a strong notion of quasiconvexity in finitely generated groups,
which we call stability. Stability agrees with quasiconvexity in hyperbolic
groups and is preserved under quasi-isometry for finitely generated groups. We
show that the stable subgroups of mapping class groups are precisely the convex
cocompact subgroups. This generalizes a well-known result of Behrstock and is
related to questions asked by Farb-Mosher and Farb.Comment: 15 pages, 1 figur
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