LALLOP

By Nicolaus Heuer and Clara Loeh. 2019

This program runs in MATLAB

*******Background

lallop computes a lower bound to the simplicial volume of one-relator group similarly to the one described in [1]. The algorithm is based on ideas of scallop [2] by Danny Calegari and implemented by Alden Walker [3].



*******Installation

Download all MATLAB files. The main computation of lallop takes place in 'lallop_arbitrary_vertex'.



*******Executing lallop

lallop has the following input:

lallop(s, l1, verbose, degree4)

Here:
-s: where we compute lallop of. It is supposed to be a string. Inverse letters are denoted by capitalization. E.g. s = 'abAB' denotes the commutator relation.

-l1: is a boolean variable. 
	-If l1=true, then lallop computes indeed lallop.
	-If l1=false, then lallop computes scl(r), using a version 			of the algorithm of Calegari-Walker.
		
-verbose: is a boolean variable. If verbose=true, then lallop returns information about the stage of the program. Else, it does not.

-degree4: is a boolean variable.
	-if degree4=false lallop computes lallop according to the 		algorithm described in [1].
	-if degree4=true lallop computes a value for lallop by just 		assuming that there are 4-pods.  This is analogous to 			'scallop' implemented by Alden Walker. This computation is 		much faster than computing lallop, and often returns the 			accurate value. However, it is not always accurate.

	



*******Examples
for s=abAB we run

lallop_arbitrary_vertex('abAB',true,false,false)

and we return 0. This makes sense as the simplicial volume is zero.

for s = abABcdCD we run

lallop_arbitrary_vertex('abABcdCD',true,false,false)

and lallop returns 4. This agrees with the simplicial volume for surface relations.

for s = abABaBBAbb (compare Example F of [1]) we run

lallop_arbitrary_vertex('abABaBBAbb',true,false,false)

and we get 0. 






*******References

[1] N. Heuer, C. Loeh, Simplicial Volume of One Relator Groups, arXiv preprint.

[2] D. Calegari. Stable commutator length is rational in free groups. J. Amer. Math. Soc. 22 (2009) no. 4.

[3] A. Walker. scallop. Computer program.