Amalgamated sums of groups

Abstract

Groups called amalgamated sums that arise as inductive limits of systems of groups and injective homomorphisms are studied. The problem is to find conditions under which the groups in the system do not collapse in the limit. Such a condition is given by J. Tits when certain subsystems are associated to buildings. This condition can be phrased to apply to certain systems of abstract groups and injective homomorphisms. It is shown to imply that no collapse occurs in the limit in a strong sense; namely the natural homomorphism of the amalgamated sum of any subsystem into the amalgamated sum of the full system is injective. This answers a question of S. J. Pride.

The Tits alternative for non-spherical triangles of groups

“…A similar result holds for non-spherical Corson diagrams, see [13]. While these two results can be proved by nice arguments based on Euler's formula for planar graphs, spherical Corson diagrams are much harder to investigate, see, for example, [2,12].…”

“…Later, it has been generalized to non-spherical Corson diagrams in [13]. Even more has been shown in [13].…”

“…This guarantees that one can still read off relators along the boundaries of the remaining vertices. Here, we leave the details to the reader, see [13,Appendix] for another description and 1 -3 in Figure 7 for some examples. Note that, in 2 , the generator ϕ J1J2 (a) · b J2 ∈ G J2 satisfies the following equation: Therefore, one can actually read off the relator a · b J1 = ϕ J1J2 (a · b J1 ) along the boundary of the respective joining vertex.…”

“…Otherwise, there is still a path from the new local vertex to D k , which implies that the new local vertex is a neighbour of some other local vertex. Once this is clear, we can remove it as in (13) and in the final step of (11). In either case, in particular in the latter, we do not only remove the new local vertex but also at least one more bridge.…”

“…Let D k be a local vertex of type J = {i, j}. By (13), D k has sufficiently many swaps in its boundary. So, the Gersten-Stallings angle {i,j} = 0 and the number of swaps is at least 2π/ {i,j} .…”