Abstract. We explore the geometry of the Cayley graphs of the lamplighter groups and a wide range of wreath products. We show that these groups have dead end elements of arbitrary depth with respect to their natural generating sets. An element w in a group G with finite generating set X is a dead end element if no geodesic ray from the identity to w in the Cayley graph Γ(G, X) can be extended past w. Additionally, we describe some nonconvex behavior of paths between elements in these Cayley graphs and seesaw words, which are potential obstructions to these graphs satisfying the kfellow traveller property.
Abstract. We discuss metric and combinatorial properties of Thompson's group T , including normal forms for elements and unique tree pair diagram representatives. We relate these properties to those of Thompson's group F when possible, and highlight combinatorial differences between the two groups. We define a set of unique normal forms for elements of T arising from minimal factorizations of elements into natural pieces. We show that the number of carets in a reduced representative of an element of T estimates the word length, that F is undistorted in T , and we describe how to recognize torsion elements in T .
Abstract. We study some combinatorial consequences of Blake Fordham's theorems on the word metric of Thompson's group F in the standard two generator presentation. We explore connections between the tree pair diagram representing an element w of F , its normal form in the infinite presentation, its word length, and minimal length representatives of it. We estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates. In addition we explore some properties of the Cayley graph of F with respect to the two generator finite presentation. Namely, we exhibit the form of "dead end" elements in this Cayley graph, and show that it has no "deep pockets". Finally, we discuss a simple method for constructing minimal length representatives for strictly positive or negative words.
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