19 research outputs found
Centers of subgroups of big mapping class groups and the Tits alternative
In this note we show that many subgroups of mapping class groups of
infinite-type surfaces without boundary have trivial centers, including all
normal subgroups. Using similar techniques, we show that every nontrivial
normal subgroup of a big mapping class group contains a nonabelian free group.
In contrast, we show that no big mapping class group satisfies the strong Tits
alternative enjoyed by finite-type mapping class groups. We also give examples
of big mapping class groups that fail to satisfy even the classical Tits
alternative and give a proof that every countable group appears as a subgroup
of some big mapping class group.Comment: 6 pages, 1 figur
Graphs of curves for surfaces with finite-invariance index (1)
In this note we make progress toward a conjecture of DurhamāFanoniāVlamis, showing that every infinite-type surface with fiĀniĀte-invariance index (1) and no nondisplaceable compact subsurfaces fails to have a good graph of curves, that is, a connected graph where vertices represent homotopy classes of essential simple closed curves and with a natural mapping class group action having infinite diameter orbits. Our arguments use tools developed by MannāRafi in their study of the coarse geometry of big mapping class groups
Centers of subgroups of big mapping class groups and the Tits alternative
In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show that no big mapping class group satisfies the strong Tits alternative enjoyed by finite-type mapping class groups. We also give examples of big mapping class groups that fail to satisfy even the classical Tits alternative; consequently, these examples are not linear
Least dilatation of pure surface braids
This thesis finds its roots in the Nielsen-Thurston classification of the mapping class group, a result that is fundamental to the field of low dimensional topology. In particular, Thurston's work gives us a powerful normal form for mapping classes: up to taking powers and restricting to subsurfaces, every mapping class can be decomposed into pieces which are either the identity or pseudo-Anosov. Associated to each of these pseudo-Anosov mapping classes is a unique algebraic number called its dilatation or ``stretch-factor". In this thesis, we build on work of Penner who introduced the study of the minimal dilatation of pseudo-Anosovs in subgroups of the mapping class group. We prove upper and lower bounds on the minimal dilatation of pseudo-Anosovs in the -stranded pure surface braid group extending results of Aougab--Taylor and Dowdall for the 1-stranded pure surface braid group
Unmarked simple length spectral rigidity for covers
We prove that every closed orientable surface S of negative Euler
characteristic admits a pair of finite-degree covers which are length
isospectral over S but generically not simple length isospectral over S. To do
this, we first characterize when two finite-degree covers of a connected,
orientable surface of negative Euler characteristic are isomorphic in terms of
which curves have simple elevations. We also construct hyperbolic surfaces X
and Y with the same full unmarked length spectrum but so that for each k, the
sets of lengths associated to curves with at most k self-intersections differ.Comment: 33 pages, 8 figures; v2 contains only the non-effective portion of
v1, the effective portion will appear in a separate postin
End-periodic homeomorphisms and volumes of mapping tori
Given an irreducible, end-periodic homeomorphism f of a surface S with
finitely many ends, all accumulated by genus, the mapping torus is the interior
of a compact, irreducible, atoroidal 3-manifold with incompressible boundary.
Our main result is an upper bound on the infimal hyperbolic volume of the
compactified mapping torus in terms of the translation length of f on the pants
graph of S. This builds on work of Brock and Agol in the finite-type setting.
We also construct a broad class of examples of irreducible, end-periodic
homeomorphisms and use them to show that our bound is asymptotically sharp.Comment: 42 pages, 9 figure