Infinite Geodesics in the Discrete Heisenberg Group

The language of geodesics for the discrete Heisenberg group

“…A right-infinite word is called geodesic if all its finite parts are geodesic. In [1] the language of infinite geodesic words in H(Z) was described explicitly. It turned out that in H(Z) there is a finite geodesic word that is not contained in any infinite one.…”

“…The representation above is said to be the geometric model of H(Z). For details see [1]. An image of geodesic words in H(Z) under the projection are oriented polygonal chains on the plane, see Figure 3.…”

“…These transformations are called geometric involutions. In Lemma 2.2. of [1] it is proved that they send geodesic words to geodesic words. It is now hard not to show that they send dead end words to dead end words.…”

Абсолют Конечно Порожденных Групп: II. Лапласова И Вырожденная Части

“…To check that this embedding is bijective, one should verify that the decomposition of a homogeneous nondegenerate central measure ν into ergodic components involves neither degenerate measures, no measures with characteristics different from that of ν. To see this, in the Cayley graph choose a cycle of nonzero length with endpoints at the identity of the group and observe that, by (5), the values of homogeneous measures with different characteristics on the sequence of powers of this cycle produce exponentials with different bases, while every degenerate measure vanishes on this sequence (see Theorem 5.1).…”

“…In this article, we continue to study the notion of the absolute of discrete groups and semigroups with a fixed system of generators, see the previous papers [1]- [3], [5], [16], [17]. It appeared as a natural generalization of the well-known notions of Poisson-Furstenberg boundary, Martin boundary, etc.…”