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Register a new userAsymptotically rigid mapping class groups I: Finiteness properties of braided Thompsons and Houghtons groups
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This article is dedicated to the study of asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy-Thompson groups $T^sharp,T^ast$ introduced by Funar and Kapoudjian, and the braided Houghton groups $mathrm{br}H_n$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube-stabilisers isomorphic to finite extensions of braid groups. As an application, we prove Funar-Kapoudjians and Degenhardts conjectures by showing that $T^sharp,T^ast$ are of type $F_infty$ and that $mathrm{br}H_n$ is of type $F_{n-1}$ but not of type $F_n$.
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This is an addendum to arXiv: 0810.5376. We show, using our methods and an auxiliary result of Bestvina-Bromberg-Fujiwara, that a finitely generated group with infinitely many pairwise non-conjugate homomorphisms to a mapping class group virtually acts non-trivially on an $R$-tree, and, if it is finitely presented, it virtually acts non-trivially on a simplicial tree
Let $Gamma$ be a finite index subgroup of the mapping class group $MCG(Sigma)$ of a closed orientable surface $Sigma$, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element $ginGamma$ has positive stable commutator length. In addition, we show that in these situations the stable commutator length, if nonzero, is uniformly bounded away from 0. The method works for certain subgroups of infinite index as well and we show $scl$ is uniformly positive on the nontrivial elements of the Torelli group. The proofs use our earlier construction in the paper Constructing group actions on quasi-trees and applications to mapping class groups of group actions on quasi-trees.
Let $G$ be a virtually special group. Then the residual finiteness growth of $G$ is at most linear. This result cannot be found by embedding $G$ into a special linear group. Indeed, the special linear group $text{SL}_k(mathbb{Z})$, for $k > 2$, has residual finiteness growth $n^{k-1}$.
We show that many 2-dimensional Artin groups are residually finite. This includes 3-generator Artin groups with labels $geq$ 3 where either at least one label is even, or at most one label is equal 3. As a first step towards residual finiteness we show that these Artin groups, and many more, split as free products with amalgamation or HNN extensions of finite rank free groups. Among others, this holds for all large type Artin groups with defining graph admitting an orientation, where each simple cycle is directed.
We show that Out(G) is residually finite if G is a one-ended group that is hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer automorphisms preserving the peripheral structure is residually finite. We also show that Out(G) is virtually p-residually finite for every prime p if G is one-ended and toral relatively hyperbolic, or infinitely-ended and virtually p-residually finite.
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