# Abstract/Details

## Artin and Dehn twist subgroups of the mapping class group

2011 2011

### Abstract (summary)

This dissertation investigates two types of subgroups in the mapping class group of an orientable surface. The first type of subgroups are isomorphic images of Artin groups. The second type of subgroups is one which is generated by three Dehn twists along simple closed curves with small geometric intersections.

Let *S* be a compact orientable surface. The mapping class group, *Mod*(*S*), of *S* is the group of isotopy classes of orientation preserving homeomorphisms of * S* fixing the boundary pointwise. *Mod*(*S*) is a very rich and complex object. In this dissertation, we make progress toward understanding the structure of the above mentioned subgroups of * Mod*(*S*).

We tackle three problems. The first problem focuses on finding embeddings of Artin groups into *Mod*(*S*). The second problem involves finding Artin relations of every length in Mod(S). And the third problem deals with understanding subgroups of *Mod*(* S*) generated by three Dehn twists along curves with small geometric intersections.

While it is easy to find nontrivial homomorphisms of Artin groups into * Mod*(*S*), the question of whether such homomorphisms are injective is quite hard. In this dissertation, we find embeddings of the Artin groups [special characters omitted](*B _{n}*), [special characters omitted](

*H*

_{3}), [special characters omitted](

*I*

_{2}(

*n*)), and most notably [special characters omitted](

*Ã*

_{n}_{−1}) into

*Mod*(

*S*). Further, we prove that if a collection {

*a*

_{1}, ···,

*a*} of simple closed curves in

_{n}*S*has curve graph (see definition 4.1.2)

*Ã*

_{n}_{−1}and

*N*

_{&epsis;}is a closed regular neighborhood of [special characters omitted]

*a*, then the subgroup of

_{i}*Mod*(

*N*) generated by the (left) Dehn twists

_{&epsis;}*T*along

_{i}*a*is isomorphic to [special characters omitted](

_{i}*Ã*

_{n}_{−1}) almost all the time.

In the second problem, we study Artin relations in the mapping class group. If *l* ≥ 2 is an integer, then *a* and *b* satisfy the Artin relation of length *l* if *aba*··· = *bab*···, where each side of the equality has *l* terms. We give explicit elements of *Mod*(*S*) satisfying Artin relations of every integer length *l* ≥ 2. By direct computations, we find elements *x* and *y* in *Mod *(*S*) satisfying Artin relations of every even length ≥ 8 and every odd length ≥ 3. Then using the theory of Artin groups, we give two methods for finding Artin relations in *Mod*(*S *). The first yields Artin relations of every length ≥ 3, while the second provides Artin relations of every even length ≥ 6. In the last two cases, we also show that *x* and *y* generate the Artin group [special characters omitted](*I*_{2}(*l*)), where * l* is the length of the Artin relation satisfied by *x* and *y*.

The third problem is concerned with understanding subgroups in * Mod*(*S*) generated by three Dehn twists along curves with small geometric intersections. Let *a*_{1}, * a*_{2}, and *a*_{3} be distinct isotopy classes of essential simple closed curves in an orientable surface * S*. Assume that *i*(*a _{j}*,

*a*) ∈ {0, 1, 2} for all

_{k}*j*,

*k.*Denote by

*T*the (left) Dehn twist along

_{i}*a*, and let

_{i}*G*represent the subgroup of

*Mod*(

*S*) generated by

*T*

_{1},

*T*

_{2}, and

*T*

_{ 3}. Set (

*x*

_{12},

*x*

_{13},

*x*

_{23}) = (

*i*(

*a*

_{1},

*a*

_{2}),

*i*(

*a*

_{1},

*a*

_{3}),

*i*(

*a*

_{2},

*a*

_{3})). We find explicit presentations for

*G*when (

*x*

_{12},

*x*

_{13},

*x*

_{23}) = (0, 0, 0), (1, 0, 0), (2, 0, 0), (1, 0, 1), and (1, 1, 1). For the triple (2, 1, 0), there are two cases to consider (see subsections 7.8.1 and 7.8.2). In both cases, we are not able to find an explicit presentation for

*G*. Nevertheless, we prove that

*G*is a subgroup of some Artin group [special characters omitted]. Moreover, using the computer algebra software Magma, we show that

*G*is finitely presented and is isomorphic to a subgroup of infinite index in [special characters omitted]. Although we have obtained similar partial results for the triples (2, 2, 0), (2, 1, 0), (2, 1, 1), (2, 2, 0), and (2, 2, 2), we do not include them in this dissertation.

While the three problems discussed above are seemingly disconnected, they are in fact intimately related. They reflect a beautiful interplay between Artin groups and mapping class groups.