On the geometry of graphs associated to infinite-type surfaces

Consider a connected orientable surface $S$ of infinite topological type, i.e. with infinitely-generated fundamental group. We describe the large-scale geometry of arbitrary connected subgraphs of the arc complex $A(S)$ and curve complex $C(S)$ of $S$, provided they are invariant under a sufficiently big subgroup of the mapping class group $Mod(S)$. We obtain a number of consequences; in particular we recover the main results of J. Bavard and Aramayona-Fossas-Parlier .Comment: v2: Substantial rewrite, fixes some errors in the previous version. Proposition 1.3 of v1 has now been merged into Theorem 1.1. Theorems 1.6 and 1.11 of v1 were not correct as stated, this has been fixed in v2. Any mention to subgraphs of the curve graph for surfaces with isolated ends was removed, as this case is covered in recent work of Durham-Fanoni-Vlami

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

Simplicial embeddings between pants graphs

We prove that, except in some low-complexity cases, every locally injective simplicial map between pants graphs is induced by a $\pi_1$-injective embedding between the corresponding surfaces.Comment: 18 pages, 3 figure

Injections of mapping class groups

We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of once-punctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multi-twists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary

Quotients of the mapping class group by power subgroups

We study the quotient of the mapping class group $\operatorname{Mod}_g^n$ of a surface of genus $g$ with $n$ punctures, by the subgroup $\operatorname{Mod}_g^n[p]$ generated by the $p$-th powers of Dehn twists. Our first main result is that $\operatorname{Mod}_g^1 /\operatorname{Mod}_g^1[p]$ contains an infinite normal subgroup of infinite index, and in particular is not commensurable to a higher-rank lattice, for all but finitely many explicit values of $p$. Next, we prove that $\operatorname{Mod}_g^0/ \operatorname{Mod}_g^0[p]$ contains a K\"ahler subgroup of finite index, for every $p\ge 2$ coprime with six. Finally, we observe that the existence of finite-index subgroups of $\operatorname{Mod}_g^0$ with infinite abelianization is equivalent to the analogous problem for $\operatorname{Mod}_g^0/ \operatorname{Mod}_g^0[p]$.Comment: 17p, Bull. London Math. Soc., to appea