The braided Ptolemy–Thompson group is finitely presented

Abstract

Pursuing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group T ] (and its companion T ) which is an extension of the Ptolemy-Thompson group T by the braid group B 1 on infinitely many strands. We prove that T ] is a finitely presented group by constructing a complex on which it acts cocompactly with finitely presented stabilizers, and derive from it an explicit presentation. The groups T ] and T are in the same relation with respect to each other as the braid groups B nC1 and B n , for infinitely many strands n. We show that both groups embed as groups of homeomorphisms of the circle and their word problem is solvable. 20F36, 57M07; 20F38, 20F05, 57N05 IntroductionThe first relationships between Thompson's groups and braid groups were brought to light in the article [25] by P Greenberg and V Sergiescu, which is devoted to the construction and the homological study of extensions of Thompson's groups F and T by the stable braid group B Thompson's groups are not tree automorphisms, but are induced by piecewise tree automorphisms [28]. Therefore, a natural question is to find a way of lifting those elements to automorphisms of an appropriate structure. The answer proposed by [28] and [22] is to lift them to mapping classes of homeomorphisms of particular surfaces. Indeed, both groups A T and B are mapping class groups of infinite surfaces which are thickenings of suitable regular trees; the surfaces are endowed with an extra structure that must be, not globally, but only asymptotically preserved by the mapping classeshence the notion of asymptotic mapping class group. This extra structure may consist of a decomposition of the surface into pairs of pants, hexagons, hexagons with punctures, and so on.The surface D ] considered for the construction of the asymptotic mapping class group T ] is the planar thickened binary tree, which is punctured along an infinite discrete subset of points. The extra structure consists of a decomposition into suitably punctured hexagons. The asymptotic mapping class group that one obtains this way is an extension T ] of T by the group of braids B 1 on infinitely many strands (corresponding to the punctures). Therefore, T ] is quite similar to, but simpler than A T .This new group T ] seems interesting and worthy of deeper study. Compared with B , the definition of T ] presents new features, for instance, the dependence on the extra structure is now clearly manifest. We can choose two sets of punctures leading to homeomorphic surfaces for which the associated groups are not isomorphic. We obtain that way another group T , which is a sort of twin brother of T ] . Although T ] and T share the same properties, they are different. Our main result is the following: T (see Penner [35]). The terminology used here for T is expected to stress on its link with the Penner-Ptolemy groupoid. The group T ] is essentially different from BV (and B ), being an extension by the whole group of braids, and not only the pure braids. Moreover, it is know...

The braided Ptolemy–Thompson group is asynchronously combable

“…An algebraic relation between T and the braid groups has been discovered in an article due to P. Greenberg and V. Sergiescu ([21]). Since then, several works ( [6], [7], [10], [11], [14], [15], [16], [28]) have contributed to improve our understanding of the links between Thompson groups and mapping class groups of surfaces -including braid groups.…”

“…The group T has received a lot of attention since E. Ghys and V. Sergiescu ([20]) proved that it embeds in the diffeomorphism group of the circle and it can be viewed as a sort of discrete analogue of the latter. The group T has been introduced in [15] as a mapping class group of an infinite surface obtained as follows. Consider first the planar surface obtained by thickening the regular binary tree, with one puncture for each edge of the tree.…”

“…The mapping classes of orientation-preserving homeomorphisms of this punctured surface, which induce a tree isomorphism outside a bounded domain, form the group T . Our main result in [15] is that T ? is finitely presented and has solvable word problem.…”

“…We proved in [15] that T is generated by two suitable lifts of the elements˛anď of T , it is finitely presented and has solvable word problem.…”

“…is the analogue of T for the punctured disk. It is not hard to see (see [15]) that one has an exact sequence…”