The distinguishing number D(G) of a graph G is the least cardinal d such that G has a labeling with d labels which is only preserved by the trivial automorphism. We show that the distinguishing number of infinite, locally finite, connected graphs G with infinite motion and growth o(n 2 / log 2 n) is either 1 or 2, which proves the Infinite Motion Conjecture of Tom Tucker for this type of graphs. The same holds true for graphs with countably many ends that do not grow too fast. We also show that graphs G of arbitrary cardinality are 2-distinguishable if every nontrivial automorphism moves at least uncountably many vertices m(G), where m(G) ≥ |Aut(G)|. This extends a result of Imrich et al. to graphs with automorphism groups of arbitrary cardinality.

We consider random walks on non-amenable Baumslag-Solitar groups BS(p, q) and describe their Poisson-Furstenberg boundary. The latter is a probabilistic model for the long-time behaviour of the random walk. In our situation, we identify it in terms of the space of ends of the Bass-Serre tree and the real line using Kaimanovich's strip criterion.1 2 BAUMSLAG-SOLITAR GROUPS a geometric boundary to BS(p, q). In Section 3, we turn to random walks on groups. We outline some results about the Poisson-Furstenberg boundary and then state Kaimanovich's strip criterion, which is an important tool to identify this boundary geometrically.In Section 4, we study random walks on BS(p, q) with finite first moment. We consider the pointwise projections of the random walk to T and H. If the random walk has negative vertical drift, then the projection to H converges almost surely to a random element in R. For the projection to T, we do not need to distinguish between different vertical drifts; as soon as 1 < p < q, it converges almost surely to a random element in ∂T. We thus endow ∂T (or even ∂T × R) with the Borel σ-algebra B ∂T (or B ∂T×R ) and the hitting measure ν ∂T (or ν ∂T×R ). Finally, Kaimanovich's strip criterion shows that the resulting probability space is isomorphic to the Poisson-Furstenberg boundary.Up to and including Section 4.1, we assume that the two non-zero integers p and q satisfy 1 ≤ p < q. Then, we restrict ourselves to the non-amenable subcase 1 < p < q. In the appendix, we explain how to obtain similar results for the remaining non-amenable cases 1 < p < −q and 1 < p = |q|. Our main result is the following. Theorem 1.1 ("identification theorem") Let Z = (Z 0 , Z 1 , . . .) be a random walk on a non-amenable Baumslag-Solitar group G = BS(p, q) with 1 < p < q and increments X 1 , X 2 , . . . of finite first moment. Depending on the vertical drift δ, we distinguish three cases:1. If δ > 0, then the Poisson-Furstenberg boundary is isomorphic to (∂T, B ∂T , ν ∂T ) endowed with the boundary map bnd ∂T : Ω → ∂T. If3. If δ = 0 and ln(A X 1 ) has finite second moment and there is an ε > 0 such that ln(1+|B X 1 |) has finite (2 + ε)-th moment, then it is isomorphic to (∂T, B ∂T , ν ∂T ) endowed with bnd ∂T : Ω → ∂T.Note that the driftless case is a little more subtle and requires additional assumptions on the moments. Here, the terms A g and B g denote the imaginary and real part of the projection of an element g ∈ G to the hyperbolic plane H. The two assumptions are certainly satisfied if X 1 has finite (2 + ε)-th moment.Further details can be found at the beginning of Section 4.1. Baumslag-Solitar groups 2.1 Amenability of Baumslag-Solitar groupsThe structure of Baumslag-Solitar groups can be studied by means of HNN extensions. Indeed, BS(p, q)is precisely the HNN extension Z * ϕ with isomorphism ϕ : pZ → qZ given by ϕ(p) := q. This allows us to use the respective machinery, such as Britton's lemma, see [Bri63], which implies that a freely reduced non-empty word w over the letters a and b and their formal inverse...

We extend the definition of the Cartesian product to graphs with loops and show that the Sabidussi-Vizing unique factorization theorem for connected finite simple graphs still holds in this context for all connected finite graphs with at least one unlooped vertex. We also prove that this factorization can be computed in O(m) time, where m is the number of edges of the given graph.

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Triangles of groups have been introduced by Gersten and Stallings. They are, roughly speaking, a generalization of the amalgamated free product of two groups and occur in the framework of Corson diagrams. First, we prove an intersection theorem for Corson diagrams. Then, we focus on triangles of groups. It has been shown by Howie and Kopteva that the colimit of a hyperbolic triangle of groups contains a non-abelian free subgroup. We give two natural conditions, each of which ensures that the colimit of a non-spherical triangle of groups either contains a non-abelian free subgroup or is virtually solvable.