I will talk about the infinite groups K_n defined by the following finite presentation:
Generators: (a,b) whenever 0 <= a < b <= n;
Relations:
(a) (a,b)(c,d)(a,b)(c,d) whenever 0 <= a < b <= c < d <= n;
(b) (a,b)(a+x,b-y)(a,b)(a+y,b-x) whenever x,y >= 0 and 0 <= a<= a+x+y < b <= n;
(c) (a,b-z)(a+y,b)(a,b-x)(a+z,b)(a,b-y)(a+x,b) whenever x,y,z > 0 and 0 <= a<= a+x+y+z = b <= n.
My interest was drawn to this group by the observation that it has some properties in common with (infinite) Coxeter groups.
In particular, it admits a faithful finite-dimensional linear representation over the real numbers.
Moreover, there is an analogue to the Tits cone.
Without the relations (c) this defines the so-called cactus group which is less interesting from this point of view.
Whereas Coxeter groups have parabolic subgroups which are again Coxeter groups, K_n has parabolic things (called residues)
which are only groupoids, each with their own Tits cone. I will give the list of finite residues which was finished
by a computer calculation by my summer project student Chris Midgley.