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Local rules for aperiodic hierarchical tilings and finitely presented nil-semigroups

Alexei Belov, Ilya Ivanov-Pogodaev

Moscow

#### Abstract.

The talk is devoted to the topological approach for hierarchical tilings. For any hierarchical locally finite graph we can construct the set of forbidden sequences of edges (words) such that any graph without these forbidden words is isomorphic to the initial hierarchical graph.

These facts correspond to Goodmann-Strauss theorem about aperiodic hierarchical tilings.

Also, this can be used as construction method for algebraic objects.

We use this method to costruct an finitely presented infinite nil-semigroup, answering on the Shevrin problem.

The method is based on considering the paths on some tiling as non trivial elements of a semigroup. Also, the structure of tiling induces the relations in the semigroup.

The tiling can be presented by the finite number of rules, so the semigroup would have the finite number of defining relations.

There are no periodic paths on the tiling so there are no periodic words in the semigroup.

The subject is related to other Burnside type problems in groups and rings.