# Injective maps between Artin groups

## Author

**John Crisp**

## Status

Research Report 97-10

Date: 27 March 1997

## Abstract

A sufficient condition is given for the injectivity of a homomorphism
between Artin monoids which moreover ensures injectivity of the
induced map between Artin groups in the case where both groups are of
finite type. We list numerous examples of monoid homomorphisms
satisfying this injectivity condition, all of which happen to be
so-called LCM-homomorphisms.
In the case of an LCM-homomorphism there is a natural way to realise
the corresponding map on Artin groups geometrically as the map
induced on fundamental groups by an inclusion of certain finite
simplicial complexes.

An interesting group homomorphism which is not realised in this way
exhibits the Artin group of type B_n as a subgroup of finite index of the
classical (n+1)-string braid group (type A_n). This subgroup is in fact
the group of n-string braids over an annulus.

## Key phrases

Artin group. braid group. monoid. Salvetti complex.

## AMS Subject Classification (1991)

Primary: 20F36

Secondary:

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Sydney Mathematics and Statistics