5 Markov compacta
The notion of Markov compactum is a generalization of the boundary of a group. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be an inverse system of finite simplicial complexes
dvipng error! exitcode was 2 (signal 0), transscript follows:
For a simplex
dvipng error! exitcode was 2 (signal 0), transscript follows:let
dvipng error! exitcode was 2 (signal 0), transscript follows:denote the inverse subsystem
dvipng error! exitcode was 2 (signal 0), transscript follows:formed by the subcomplexes ( building blocks)
dvipng error! exitcode was 2 (signal 0), transscript follows:
The inverse system
dvipng error! exitcode was 2 (signal 0), transscript follows:is called Markov if it contains only finitely many isomorphism classes of inverse subsystems
dvipng error! exitcode was 2 (signal 0), transscript follows:. A Markov compactum is a compactum obtained as the inverse limit of a Markov inverse system.
Thus
dvipng error! exitcode was 2 (signal 0), transscript follows:is obtained from
dvipng error! exitcode was 2 (signal 0), transscript follows:by inductively replacing simplices
dvipng error! exitcode was 2 (signal 0), transscript follows:in
dvipng error! exitcode was 2 (signal 0), transscript follows:with the building blocks
dvipng error! exitcode was 2 (signal 0), transscript follows:using only finitely many "replacement rules". For instance, the Pontryagin surfaces are Markov compacta. Markov compacta appear naturally as boundaries of hyperbolic and Coxeter groups.
For every compactum
dvipng error! exitcode was 2 (signal 0), transscript follows:either
dvipng error! exitcode was 2 (signal 0), transscript follows:
or
dvipng error! exitcode was 2 (signal 0), transscript follows:
In the latter case,
dvipng error! exitcode was 2 (signal 0), transscript follows:is called a Boltyansky compactum.
Problem 5.1 (Comments) [Alexander Dranishnikov] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a compactum which is a Z-boundary of a group
dvipng error! exitcode was 2 (signal 0), transscript follows:. Then
dvipng error! exitcode was 2 (signal 0), transscript follows:is never a Boltyansky compactum.
In the special case when
dvipng error! exitcode was 2 (signal 0), transscript follows:is an Markov compactum, so that all building blocks
dvipng error! exitcode was 2 (signal 0), transscript follows:
are isomorphic, it was proven in MR2310467 that
dvipng error! exitcode was 2 (signal 0), transscript follows:cannot be a Boltyansky compactum.