We present an algorithm to decide whether or not a finitely generated subgroup of the isometry group locally finite simplicial tree is both discrete and free. The correctness this relies on following conjecture: every ‘minimal’ n-tuple isometries either contains elliptic element satisfies hypotheses Ping Pong Lemma. prove conjecture for n=2,3, show that it implies generalisation Ihara's Theorem.

We discuss the approximability of locally compact groups by finite semigroups and finite quasigroups (latin squares). We show that if a locally compact group G is approximable by finite semigroups, then it is approximable by finite groups, and thus many important groups are not approximable by finite semigroups. This result implies, in particular, the impossibility to simulate the field of real...