Hyperbolic Triangular Buildings Without Periodic Planes of Genus 2

Abstract

We study surface subgroups of groups acting simply transitively on vertex sets of certain hyperbolic triangular buildings. The study is motivated by Gromov's famous surface subgroup question: Does every one-ended hyperbolic group contain a subgroup which is isomorphic to the fundamental group of a closed surface of genus at least 2? In [9] and [3] the authors constructed and classified all groups acting simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. These groups gave the first examples of cocompact lattices acting simply transitively on vertices of hyperbolic triangular Kac-Moody buildings that are not right-angled. Here we study surface subgroups of the 23 torsion free groups obtained in [9]. With the help of computer searches we show, that in most of the cases there are no periodic apartments invariant under the action of a genus two surface. The existence of such an action implies the existence of a surface subgroup, but it is not known, whether the existence of a surface subgroup implies the existence of a periodic apartment. These groups are the first candidates for groups that have no surface subgroups arising from periodic apartments.

Finding Periodic Apartments via Boolean Satisfiability and Orderly Generation

“…In terms of GSC, we are able to rule out further 4 groups from the remaining set of possible counterexamples to GSC. Our results also serve as an independent validation of the results in [25] for g = 2.…”

“…For scaling to g = 4, we apply a combination of orderly generation of the graphs and their cyclesets (obtaining strong symmetry breaking), and employ SAT solving for the final labeling task. As whole, this combination of techniques provides massive scalability improvements over the specialized approach of [25]. In terms of GSC, we are able to rule out further 4 groups from the remaining set of possible counterexamples to GSC.…”

“…Within this class of hyperbolic groups, a set of 23 counterexample candidates were recently identified [24]. As shown in [25], the question of whether a particular one of these groups contains a surface subgroup arising from so-called periodic apartments is equivalent to determining whether a specific type of graph (with a non-trivial combination of properties) exists. Of interest from the view point of computational mathematics is the fact that ruling each individual group of the 23 out as counterexamples to GSC boils down to determining whether there exist bipartite, 3-regular, connected multigraphs that both decompose exactly into sets of cycles of specific lengths and admit a type of a labeling using a representation of the group in question [25].…”

“…In the rest of this paper, we will refrain from requiring in-depth background on geometric group theory from the reader, and will focus on the task of determining whether small graphs with the just-mentioned properties exist. In particular, as first recently considered in [25], this graph search problem admits computational approaches towards ruling out counterexamples candidates to GSC. The size of the graphs of interest is controlled by a parameter called the genus, arising from the classification of surfaces in topology.…”

“…The size of the graphs of interest is controlled by a parameter called the genus, arising from the classification of surfaces in topology. Towards ruling out the 23 groups as potential counterexamples to GSC, a specialized computational approach was developed in [25] for exhaustively searching over all graphs under the smallest genus value g = 2. The approach consisted of generating all of the bipartite, 3-regular multigraphs for g = 2, with 16(g − 1) = 16 nodes and 24(g − 1) = 24 edges, and performing a specialized depth-first search for each of the graphs.…”